This document contains information about trigonometry and using the tangent ratio (tan) to solve problems in right triangles. It includes:
1) An introduction to trigonometry and defining the adjacent, opposite, and hypotenuse sides of a right triangle.
2) Examples of using tan to calculate angles and side lengths, such as finding the height of a tower given the opposite side length and angle.
3) Real-life examples like calculating the height of an airplane above the ground given the distance from the airport and angle of descent.
This document provides information about trigonometry and using trigonometric ratios like tangent, sine and cosine to solve problems involving right-angled triangles. It includes examples of using the tangent ratio to calculate heights and lengths, finding missing angles and sides. Real-life examples involve calculating heights of trees and planes. Instructions are provided on using a calculator to evaluate trigonometric functions. Exercises are included to solve problems using tangent, sine and other ratios.
1. The document discusses trigonometric ratios and how to use them to solve for missing side lengths and angle measures in right triangles.
2. It provides examples of setting up trig ratios, using the Pythagorean theorem, and using inverse trig functions to find missing angles.
3. The key steps are to label the sides of the right triangle, set up the appropriate trig ratios based on which information is known or missing, and use trig identities or the inverse functions to calculate the missing information.
This document discusses solving right triangle problems using trigonometric functions like sine, cosine, and tangent. It provides examples of right triangles where certain sides or angles are given and asks the reader to solve for unknown values. Applications discussed include finding the length of a pole or distance across a river using right triangles defined by angles of elevation or depression. Bearings and courses of ships are also defined and used in an example of finding distance and bearings between two moving ships after a period of time.
This document discusses trigonometric ratios and functions. It defines the sine, cosine, and tangent ratios for right triangles. It also introduces the reciprocal functions of cosecant, secant, and cotangent. Special angle values for trig functions are provided in a table. The document explains how to write trig functions in terms of x, y, and r using the Cartesian plane and use a CAST diagram to determine the quadrants where trig ratios are positive and negative. Finally, it discusses extending knowledge of special angles using trig ratio definitions in the Cartesian plane.
This document provides examples and explanations of right triangle trigonometry concepts, including:
1) Finding trigonometric ratios (sine, cosine, tangent) for given angles in right triangles.
2) Using trigonometric functions to find missing side lengths, including for special 30-60-90 and 45-45-90 right triangles.
3) Applying trigonometry to real-world problems involving angles of elevation/depression and finding distances.
Reciprocal trig functions (cosecant, secant, cotangent) and the Pythagorean theorem are also discussed. Practice problems are provided to test understanding.
The document discusses the Pythagorean theorem and its applications to right triangles. It defines the Pythagorean theorem, which states that the sum of the squares of the legs of a right triangle equals the square of the hypotenuse. It provides examples of using the Pythagorean theorem to calculate missing sides of right triangles when given two sides. Exercises at the end ask the reader to apply the Pythagorean theorem to find missing sides of right triangles.
1) The document outlines an assignment and lesson on parallelograms. It includes examples of finding the perimeter, area, and angle measures of various parallelograms.
2) The warm-up questions cover evaluating expressions, calculating sales tax, and converting fractions to decimals and percents.
3) The lesson defines the formula for finding the area of a parallelogram and lists properties of parallelograms including opposite angles being congruent and adjacent angles summing to 180 degrees.
The document provides formulas and definitions for various topics in geometry, algebra, trigonometry, and other areas of mathematics. Some of the key information presented includes formulas for calculating the volume and surface area of geometric solids like cubes, cylinders, and cones. In algebra, formulas are given for exponent rules, factoring, and solving equations. Trigonometry formulas define the trigonometric functions and include basic identities. Graphs of common functions like linear, quadratic, and trigonometric functions are also depicted.
This document provides information about trigonometry and using trigonometric ratios like tangent, sine and cosine to solve problems involving right-angled triangles. It includes examples of using the tangent ratio to calculate heights and lengths, finding missing angles and sides. Real-life examples involve calculating heights of trees and planes. Instructions are provided on using a calculator to evaluate trigonometric functions. Exercises are included to solve problems using tangent, sine and other ratios.
1. The document discusses trigonometric ratios and how to use them to solve for missing side lengths and angle measures in right triangles.
2. It provides examples of setting up trig ratios, using the Pythagorean theorem, and using inverse trig functions to find missing angles.
3. The key steps are to label the sides of the right triangle, set up the appropriate trig ratios based on which information is known or missing, and use trig identities or the inverse functions to calculate the missing information.
This document discusses solving right triangle problems using trigonometric functions like sine, cosine, and tangent. It provides examples of right triangles where certain sides or angles are given and asks the reader to solve for unknown values. Applications discussed include finding the length of a pole or distance across a river using right triangles defined by angles of elevation or depression. Bearings and courses of ships are also defined and used in an example of finding distance and bearings between two moving ships after a period of time.
This document discusses trigonometric ratios and functions. It defines the sine, cosine, and tangent ratios for right triangles. It also introduces the reciprocal functions of cosecant, secant, and cotangent. Special angle values for trig functions are provided in a table. The document explains how to write trig functions in terms of x, y, and r using the Cartesian plane and use a CAST diagram to determine the quadrants where trig ratios are positive and negative. Finally, it discusses extending knowledge of special angles using trig ratio definitions in the Cartesian plane.
This document provides examples and explanations of right triangle trigonometry concepts, including:
1) Finding trigonometric ratios (sine, cosine, tangent) for given angles in right triangles.
2) Using trigonometric functions to find missing side lengths, including for special 30-60-90 and 45-45-90 right triangles.
3) Applying trigonometry to real-world problems involving angles of elevation/depression and finding distances.
Reciprocal trig functions (cosecant, secant, cotangent) and the Pythagorean theorem are also discussed. Practice problems are provided to test understanding.
The document discusses the Pythagorean theorem and its applications to right triangles. It defines the Pythagorean theorem, which states that the sum of the squares of the legs of a right triangle equals the square of the hypotenuse. It provides examples of using the Pythagorean theorem to calculate missing sides of right triangles when given two sides. Exercises at the end ask the reader to apply the Pythagorean theorem to find missing sides of right triangles.
1) The document outlines an assignment and lesson on parallelograms. It includes examples of finding the perimeter, area, and angle measures of various parallelograms.
2) The warm-up questions cover evaluating expressions, calculating sales tax, and converting fractions to decimals and percents.
3) The lesson defines the formula for finding the area of a parallelogram and lists properties of parallelograms including opposite angles being congruent and adjacent angles summing to 180 degrees.
The document provides formulas and definitions for various topics in geometry, algebra, trigonometry, and other areas of mathematics. Some of the key information presented includes formulas for calculating the volume and surface area of geometric solids like cubes, cylinders, and cones. In algebra, formulas are given for exponent rules, factoring, and solving equations. Trigonometry formulas define the trigonometric functions and include basic identities. Graphs of common functions like linear, quadratic, and trigonometric functions are also depicted.
Trigonometric functions in standard position slide 1Jessica Garcia
This document discusses trigonometric functions of angles in standard position. It defines standard position as having the angle's vertex at the origin with one ray on the positive x-axis. Examples are provided for finding the reference angle of given angles and calculating trigonometric functions based on the coordinates of a point on the terminal side of the angle. Practice problems are included to draw angles in standard position and find trig values based on given angle measures or point coordinates.
The document discusses trigonometry and using a scientific calculator to solve trigonometric problems involving right triangles. It defines the sides of a right triangle as the hypotenuse, opposite, and adjacent sides. It explains how to use the sine, cosine, and tangent functions on a calculator to find missing angles and sides of a triangle when given other information. Examples are provided for finding unknown angles from side lengths and unknown sides when given an angle measure. The document also addresses situations where the unknown value is on the bottom of a trigonometric ratio fraction.
The document provides information about trigonometry topics for a foundation level class including: solving problems using right triangles, the Pythagorean theorem, defining trig functions like sine, cosine, and tangent for all angle values, graphing periodic trig functions like sine and cosine waves, and noting that the tangent graph has periods of 180 degrees and vertical asymptotes every 90 degrees.
This document discusses trigonometric functions of angles. It defines the four quadrants formed by the x and y axes and explains how the quadrant an angle lies in determines the signs of its trigonometric functions. Quadrantal angles which lie on the x or y axes have easily calculable trig functions. Any angle's reference angle is the angle between its terminal side and the nearest x-axis, allowing the calculation of trig functions without a calculator. Examples are provided to demonstrate finding trig functions based on an angle's quadrant or using its reference angle.
1) The document provides instructions for homework assignments and an upcoming test. It also includes warm-up problems and lessons on finding the perimeter and area of parallelograms, as well as determining unknown angle measures in parallelograms using properties such as opposite angles being congruent.
2) Examples are given for finding the perimeter and area of specific parallelograms. Additional problems require determining angle measures using given information about parallelograms.
3) The final problems involve finding unknown angle measures in a parallelogram figure where some angle measures are given.
This document discusses trigonometry and its applications to triangles. It introduces the key trigonometric functions of sine, cosine, and tangent using the ratios of sides in right triangles. These ratios are summarized by the mnemonic SOH CAH TOA. Examples are provided to demonstrate using trigonometric functions to find unknown side lengths, angles, and areas of triangles. Non-right triangles are also discussed along with example story problems involving trigonometry.
Trigonometry deals with relationships between the sides and angles of triangles. It uses trigonometric functions like sine, cosine, and tangent. Sine relates the opposite side to the hypotenuse, cosine relates the adjacent side to the hypotenuse, and tangent relates the opposite side to the adjacent side. Trigonometry can be used to find unknown side lengths or angles in triangles, including solving "story problems" where trigonometric applications are needed to determine real-world measurements. The document provides examples of using trigonometric functions like sine, cosine, and tangent to solve for unknown values in right and non-right triangles, as well as how to set up and solve word problems involving trigonometric applications.
This document introduces right triangle trigonometry. It defines the six trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant) using the sides of a right triangle. Examples are provided to evaluate the trig functions of given angles and to solve application problems involving right triangles. Special right triangles and trigonometric identities are also discussed. Students are assigned homework problems evaluating trig functions and solving right triangle applications, including using a calculator.
Chapter 3 Polygons
3.1 Definition
3.2 Terminology
3.3 Sum Of Interior Angles Of A Polygon
3.4 Sum Of Exterior Angles Of A Polygon
3.5 Diagonals in one vertex of any Polygon
3.6 Diagonals in any vertices of a Polygon
3.7 Quadrilaterals
Area of focus: Trigonometry and mathematical proofs
Topics covered:
> Trigonometry
> Right triangle definitions
> Trigonometric functions
> Special right triangles
> Law of sines
> Law of cosines
> Postulates and axioms
> Theorems
> Pythagorean Theorem
> Mathematical proof
Suggested time to complete (2 hrs):
> Teaching material (40 minutes)
> Practice activity (20 minutes)
> Final project (60 minutes)
This document provides instruction on right triangles and trigonometric ratios. It begins with examples of finding missing angles and side lengths in right triangles using trigonometric functions like sine, cosine and tangent. Special right triangles involving 30-60-90 and 45-45-90 triangles are discussed. Real world applications include finding the height a skier leaves a ramp and the height of a tree using an angle of elevation. The document also covers cosecant, secant and cotangent functions and has practice problems for students.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The cosine law can be used to find missing side lengths or angles in any triangle where two sides and the angle between them are known. The sine law relates the sines of the angles of a triangle to the lengths of the sides opposite them.
The document explains the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. It provides examples of using the theorem to calculate the length of the hypotenuse or legs of a right triangle when the other two sides are known. These include using the theorem to calculate the distance between two points if the distances travelled in each direction are known.
The document provides instruction on completing the square, a method for solving quadratic equations that are not factorable. It explains the five steps: 1) divide by the leading coefficient, 2) rewrite in the form ax + by = c, 3) find half of b and square it, adding it to both sides, 4) rewrite the perfect square trinomial and take the square root of both sides, and 5) solve for x. An example is worked through applying these steps to solve the equation x^2 - 16x + 15 = 0. The document also reviews graphing quadratic functions and finding solutions, axis of symmetry, vertex and y-intercepts.
The document explains the Pythagorean theorem and how to use it to determine if a triangle is a right triangle. It provides the formula, A2 + B2 = C2, where C is the hypotenuse and A and B are the legs. It then works through examples of applying the formula to find the length of an unknown side of different right triangles. Finally, it explains the converse of the theorem, which is that if the sum of the squares of two sides equals the square of the third side, then the triangle must be a right triangle.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Pythagoras, a Greek mathematician, is credited with discovering this theorem. The theorem can be used to calculate an unknown side of a right triangle if the other two sides are known. Worked examples are provided to demonstrate finding an unknown hypotenuse or leg using the Pythagorean theorem.
This document discusses trigonometric functions and their applications. It defines the six trigonometric functions - sine, cosine, tangent, cotangent, secant, and cosecant - using ratios of sides of a right triangle. Examples are provided to evaluate the trig functions of given angles and to use identities to relate functions. The document also discusses applications of solving right triangles by using trig functions when given angle and side length information.
The document discusses the signs of trigonometric functions based on the quadrant of the angle θ. It states that all trig functions are positive in quadrant I, and have either a negative x-coordinate and positive y-coordinate (quadrant II), negative x and y coordinates (quadrant III), or positive x and negative y (quadrant IV). It also lists the sign relationships for the trig functions when θ is replaced with -θ.
This document provides information about trigonometry and using trigonometric ratios to solve problems involving right-angled triangles. It includes examples of using the tangent, sine and inverse tangent/sine ratios to calculate missing angles and lengths. Students are encouraged to practice similar examples and exercises from their textbook to reinforce using trigonometry in real-life scenarios like calculating heights and distances. Various diagrams illustrate applying trigonometric concepts like SOHCAHTOA to solve problems step-by-step.
This document provides an overview of trigonometric applications and solving triangles using trigonometric ratios. It includes definitions of important trigonometric concepts like the triangle sum theorem, Pythagorean theorem, inverse trig functions, and solving various right triangle applications involving angles of elevation/depression. Examples are provided to demonstrate solving for missing angles and sides of triangles, as well as real world application problems involving ladders, flagpoles, and plane descent angles.
This document provides a review of exercises for a Math 112 final exam. It contains 31 multi-part exercises covering topics like graphing, logarithms, trigonometry, and word problems. The review is intended to help students practice problems similar to what may appear on the exam. The exam will have two parts, one allowing a calculator and one not.
This document provides an overview of trigonometry of right-angled triangles. It contains three main points:
1. It outlines objectives of finding unknown sides and angles in right-angled triangles using trigonometric ratios and inverse trigonometric functions.
2. It advertises an online resource for teaching videos, topic tests, and interactive practice questions on trigonometry organized by topic and difficulty.
3. It shares a personal anecdote of how the author first learned trigonometry and was able to apply it to write a computer program to draw an analogue clock.
Trigonometric functions in standard position slide 1Jessica Garcia
This document discusses trigonometric functions of angles in standard position. It defines standard position as having the angle's vertex at the origin with one ray on the positive x-axis. Examples are provided for finding the reference angle of given angles and calculating trigonometric functions based on the coordinates of a point on the terminal side of the angle. Practice problems are included to draw angles in standard position and find trig values based on given angle measures or point coordinates.
The document discusses trigonometry and using a scientific calculator to solve trigonometric problems involving right triangles. It defines the sides of a right triangle as the hypotenuse, opposite, and adjacent sides. It explains how to use the sine, cosine, and tangent functions on a calculator to find missing angles and sides of a triangle when given other information. Examples are provided for finding unknown angles from side lengths and unknown sides when given an angle measure. The document also addresses situations where the unknown value is on the bottom of a trigonometric ratio fraction.
The document provides information about trigonometry topics for a foundation level class including: solving problems using right triangles, the Pythagorean theorem, defining trig functions like sine, cosine, and tangent for all angle values, graphing periodic trig functions like sine and cosine waves, and noting that the tangent graph has periods of 180 degrees and vertical asymptotes every 90 degrees.
This document discusses trigonometric functions of angles. It defines the four quadrants formed by the x and y axes and explains how the quadrant an angle lies in determines the signs of its trigonometric functions. Quadrantal angles which lie on the x or y axes have easily calculable trig functions. Any angle's reference angle is the angle between its terminal side and the nearest x-axis, allowing the calculation of trig functions without a calculator. Examples are provided to demonstrate finding trig functions based on an angle's quadrant or using its reference angle.
1) The document provides instructions for homework assignments and an upcoming test. It also includes warm-up problems and lessons on finding the perimeter and area of parallelograms, as well as determining unknown angle measures in parallelograms using properties such as opposite angles being congruent.
2) Examples are given for finding the perimeter and area of specific parallelograms. Additional problems require determining angle measures using given information about parallelograms.
3) The final problems involve finding unknown angle measures in a parallelogram figure where some angle measures are given.
This document discusses trigonometry and its applications to triangles. It introduces the key trigonometric functions of sine, cosine, and tangent using the ratios of sides in right triangles. These ratios are summarized by the mnemonic SOH CAH TOA. Examples are provided to demonstrate using trigonometric functions to find unknown side lengths, angles, and areas of triangles. Non-right triangles are also discussed along with example story problems involving trigonometry.
Trigonometry deals with relationships between the sides and angles of triangles. It uses trigonometric functions like sine, cosine, and tangent. Sine relates the opposite side to the hypotenuse, cosine relates the adjacent side to the hypotenuse, and tangent relates the opposite side to the adjacent side. Trigonometry can be used to find unknown side lengths or angles in triangles, including solving "story problems" where trigonometric applications are needed to determine real-world measurements. The document provides examples of using trigonometric functions like sine, cosine, and tangent to solve for unknown values in right and non-right triangles, as well as how to set up and solve word problems involving trigonometric applications.
This document introduces right triangle trigonometry. It defines the six trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant) using the sides of a right triangle. Examples are provided to evaluate the trig functions of given angles and to solve application problems involving right triangles. Special right triangles and trigonometric identities are also discussed. Students are assigned homework problems evaluating trig functions and solving right triangle applications, including using a calculator.
Chapter 3 Polygons
3.1 Definition
3.2 Terminology
3.3 Sum Of Interior Angles Of A Polygon
3.4 Sum Of Exterior Angles Of A Polygon
3.5 Diagonals in one vertex of any Polygon
3.6 Diagonals in any vertices of a Polygon
3.7 Quadrilaterals
Area of focus: Trigonometry and mathematical proofs
Topics covered:
> Trigonometry
> Right triangle definitions
> Trigonometric functions
> Special right triangles
> Law of sines
> Law of cosines
> Postulates and axioms
> Theorems
> Pythagorean Theorem
> Mathematical proof
Suggested time to complete (2 hrs):
> Teaching material (40 minutes)
> Practice activity (20 minutes)
> Final project (60 minutes)
This document provides instruction on right triangles and trigonometric ratios. It begins with examples of finding missing angles and side lengths in right triangles using trigonometric functions like sine, cosine and tangent. Special right triangles involving 30-60-90 and 45-45-90 triangles are discussed. Real world applications include finding the height a skier leaves a ramp and the height of a tree using an angle of elevation. The document also covers cosecant, secant and cotangent functions and has practice problems for students.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The cosine law can be used to find missing side lengths or angles in any triangle where two sides and the angle between them are known. The sine law relates the sines of the angles of a triangle to the lengths of the sides opposite them.
The document explains the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. It provides examples of using the theorem to calculate the length of the hypotenuse or legs of a right triangle when the other two sides are known. These include using the theorem to calculate the distance between two points if the distances travelled in each direction are known.
The document provides instruction on completing the square, a method for solving quadratic equations that are not factorable. It explains the five steps: 1) divide by the leading coefficient, 2) rewrite in the form ax + by = c, 3) find half of b and square it, adding it to both sides, 4) rewrite the perfect square trinomial and take the square root of both sides, and 5) solve for x. An example is worked through applying these steps to solve the equation x^2 - 16x + 15 = 0. The document also reviews graphing quadratic functions and finding solutions, axis of symmetry, vertex and y-intercepts.
The document explains the Pythagorean theorem and how to use it to determine if a triangle is a right triangle. It provides the formula, A2 + B2 = C2, where C is the hypotenuse and A and B are the legs. It then works through examples of applying the formula to find the length of an unknown side of different right triangles. Finally, it explains the converse of the theorem, which is that if the sum of the squares of two sides equals the square of the third side, then the triangle must be a right triangle.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Pythagoras, a Greek mathematician, is credited with discovering this theorem. The theorem can be used to calculate an unknown side of a right triangle if the other two sides are known. Worked examples are provided to demonstrate finding an unknown hypotenuse or leg using the Pythagorean theorem.
This document discusses trigonometric functions and their applications. It defines the six trigonometric functions - sine, cosine, tangent, cotangent, secant, and cosecant - using ratios of sides of a right triangle. Examples are provided to evaluate the trig functions of given angles and to use identities to relate functions. The document also discusses applications of solving right triangles by using trig functions when given angle and side length information.
The document discusses the signs of trigonometric functions based on the quadrant of the angle θ. It states that all trig functions are positive in quadrant I, and have either a negative x-coordinate and positive y-coordinate (quadrant II), negative x and y coordinates (quadrant III), or positive x and negative y (quadrant IV). It also lists the sign relationships for the trig functions when θ is replaced with -θ.
This document provides information about trigonometry and using trigonometric ratios to solve problems involving right-angled triangles. It includes examples of using the tangent, sine and inverse tangent/sine ratios to calculate missing angles and lengths. Students are encouraged to practice similar examples and exercises from their textbook to reinforce using trigonometry in real-life scenarios like calculating heights and distances. Various diagrams illustrate applying trigonometric concepts like SOHCAHTOA to solve problems step-by-step.
This document provides an overview of trigonometric applications and solving triangles using trigonometric ratios. It includes definitions of important trigonometric concepts like the triangle sum theorem, Pythagorean theorem, inverse trig functions, and solving various right triangle applications involving angles of elevation/depression. Examples are provided to demonstrate solving for missing angles and sides of triangles, as well as real world application problems involving ladders, flagpoles, and plane descent angles.
This document provides a review of exercises for a Math 112 final exam. It contains 31 multi-part exercises covering topics like graphing, logarithms, trigonometry, and word problems. The review is intended to help students practice problems similar to what may appear on the exam. The exam will have two parts, one allowing a calculator and one not.
This document provides an overview of trigonometry of right-angled triangles. It contains three main points:
1. It outlines objectives of finding unknown sides and angles in right-angled triangles using trigonometric ratios and inverse trigonometric functions.
2. It advertises an online resource for teaching videos, topic tests, and interactive practice questions on trigonometry organized by topic and difficulty.
3. It shares a personal anecdote of how the author first learned trigonometry and was able to apply it to write a computer program to draw an analogue clock.
The document provides information about trigonometric ratios including sine, cosine, and tangent ratios. It gives examples of using each ratio to calculate lengths and angles in right triangles. It also provides mixed problems examples that require using more than one ratio to solve.
This document discusses direct proportion and methods for solving direct proportion problems. Direct proportion exists when two quantities change at a constant rate with respect to each other. The cross-multiplication method can be used to solve direct proportion problems by setting up a proportion between the known quantities and cross-multiplying to solve for the unknown quantity. Graphs of direct proportion relationships will always produce a straight line passing through the origin.
1) The document is a formula sheet for quantitative ability and entrance exams that provides formulas and properties for arithmetic, algebra, geometry, trigonometry, and statistics.
2) It includes tables, rules, and definitions for topics like percentages, fractions, logarithms, progressions, roots, factoring, binomials, counting principles, and probability.
3) The formula sheet is intended to be a one-stop reference for various mathematical concepts tested in exams like the CAT.
1) The document is a formula sheet for quantitative ability topics for CAT and management entrance tests provided by the website snapwiz.co.in.
2) It includes formulas and properties for arithmetic, percentages, fractions, logarithms, progressions, roots of quadratic equations, counting principles, probability, geometry, triangles, polygons, and circles.
3) Visitors to the website can access free mock CAT tests and other resources after reviewing this formula sheet.
The document discusses trigonometric ratios and right triangles. It defines sine, cosine, and tangent ratios and explains how to calculate them using the sides of a right triangle. It also discusses trigonometric functions on a coordinate plane and how to find trig ratios for special triangles like 45-45-90 triangles or using a calculator for other angles. Mnemonics for remembering the SOH CAH TOA method are provided. Examples of solving problems using trig ratios are included.
The document discusses trigonometric ratios and right triangles. It defines sine, cosine, and tangent ratios and explains how to calculate them using the sides of a right triangle. It also discusses trigonometric functions on a coordinate plane and how to find trig ratios for special triangles like 45-45-90 triangles or using a calculator for other angles. Mnemonics for remembering the SOH CAH TOA method are provided. Examples of solving problems using trig ratios are included.
The document defines trigonometric functions using ratios of sides of a right triangle. It gives examples of calculating trig functions for 45, 30, and 60 degree angles using special right triangles. It also covers trigonometric identities, including cofunction, reciprocal, and Pythagorean identities. Examples are provided to calculate trig functions given one function value.
This document provides an overview of trigonometric identities and examples of their applications. It discusses basic identities, verifying identities, and advanced identities involving sums, differences, doubles angles, and half angles. Examples are provided for applying product-to-sum and sum-to-product identities. The objectives are to review and apply various trigonometric identities to simplify expressions and evaluate functions.
This document contains notes from a calculus workshop covering several topics:
1) Arc length and applications of integrals.
2) Probability density functions and using integrals to find probabilities and means.
3) Parametric equations and eliminating parameters to sketch curves.
4) Vectors, dot products, cross products, and using them to find angles between vectors.
5) Coordinate systems including Cartesian, polar, cylindrical and spherical coordinates.
6) Double and triple integrals including finding areas, volumes, and changing coordinates.
Lecture 14 section 5.3 trig fcts of any anglenjit-ronbrown
This document discusses trigonometric functions and the unit circle approach. It defines the six trigonometric functions using points on the unit circle, where the radius is 1. Special right triangles like the 45-45-90 and 30-60-90 triangles are used to determine exact trigonometric function values for angles of 45°, 30°, and 60°. Reference angles are defined as the acute angle between the terminal side of an angle and the x-axis, and are used to determine trigonometric function values for angles in any quadrant. Examples are provided to demonstrate finding trig values using reference angles.
AS LEVEL Trigonometry (CIE) EXPLAINED WITH EXAMPLE AND DIAGRAMSRACSOelimu
This document discusses trigonometry concepts including terminology, trigonometric graphs, and using trigonometric functions. It introduces key terms like amplitude, wavelength, period, and transformations of trigonometric graphs by changing variables a, b, and d in the general equation y=a*sin(bx)+d. It also covers using trigonometric functions to find the values of sine, cosine, and tangent at special angles like 45, 60, 90 degrees. Finally, it discusses finding the reference angle and inverse of trigonometric relations.
This document contains a collection of math problems, definitions, explanations and examples related to various math topics. It includes 23 sections with problems on fractions, ratios, proportions, sequences, word problems and more. Key concepts explained include direct and inverse proportions, prime numbers, factors and divisors, operations with fractions and decimals. Several math problems are presented without solutions for practice, such as finding the width of strips to cut cloth or calculating travel times based on rates.
This document provides instruction on solving right triangles. It begins by defining the parts of a right triangle and explaining that to solve one, you need either two side lengths or one side length and one acute angle measure. Examples are worked through to demonstrate using trigonometric ratios like sine, cosine and tangent to find missing side lengths or angle measures. The document also applies right triangle concepts to real-life problems involving the glide angle of a space shuttle. Upcoming assignments are outlined, including a chapter 9 review, test and vectors lesson.
This document contains a 10 question review for a trigonometry midterm examination. It reviews concepts like using trigonometric functions to solve for unknown sides and angles of right triangles, evaluating trig functions for special angles, finding arc lengths and sector areas of circles, graphing trig functions involving amplitude, period and phase shift, and using trigonometric identities. The problems are worked out step-by-step showing the reasoning and math steps to arrive at the solutions, often leaving final answers in radical or pi form.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
3. 20 Aug 201720 Aug 2017 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
www.mathsrevision.com
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
2.2. Work out Tan Ratio.Work out Tan Ratio.
1. To identify the
hypotenuse, opposite and
adjacent sides in a right
angled triangle.
Angles & TrianglesAngles & Triangles
1.1. Understand the termsUnderstand the terms
hypotenuse, opposite andhypotenuse, opposite and
adjacent in right angledadjacent in right angled
triangle.triangle.
9. www.mathsrevision.com
TrigonometryTrigonometry
Tan 25Tan 25°° 0.4660.466
Tan 26Tan 26°° 0.4880.488
Tan 27Tan 27°° 0.5100.510
Tan 28Tan 28°° 0.5320.532
Tan 29Tan 29°° 0.5540.554
Tan 30Tan 30°° 0.5770.577
Tan 31Tan 31°° 0.6010.601
Tan 32Tan 32°° 0.6250.625
Tan 33Tan 33°° 0.6490.649
Tan 34Tan 34°° 0.6750.675
Tan 30° = 0.577
Accurate to
3 decimal places!
The ancient Greeks
discovered this and
repeated this for
all possible angles.
S3
Credit
20. 20 Aug 201720 Aug 2017 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
www.mathsrevision.com
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
2.2. Use tan of an angle to solveUse tan of an angle to solve
problems.problems.
1. To use tan of the angle to
solve problems.
Angles & TrianglesAngles & Triangles
1.1. Write down tan ratio.Write down tan ratio.
28. 20 Aug 201720 Aug 2017 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
www.mathsrevision.com
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
2.2. Use tan of an angle to solveUse tan of an angle to solve
REAL LIFE problems.REAL LIFE problems.
1. To use tan of the angle to
solve REAL LIFE problems.
Angles & TrianglesAngles & Triangles
1.1. Write down tan ratio.Write down tan ratio.
29. www.mathsrevision.com
TrigonometryTrigonometryS3
Credit
SO
H CA
H TO
A
20 Aug 201720 Aug 2017 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Use the tan ratio to find the height h of the tree
to 2 decimal places.
47o
8m
rod
o opp h
tan 47 = =
adj 8
o h
tan 47 =
8
o
h = 8× tan 47
h = 8.58m
30. www.mathsrevision.com
TrigonometryTrigonometryS3
Credit
6o
20 Aug 2017
Compiled by Mr. Lafferty Maths
Dept.
Aeroplane
a = 15
c
LennoxtownAirport
Q1.Q1. An aeroplane is preparing to land at Glasgow Airport.An aeroplane is preparing to land at Glasgow Airport.
It is over Lennoxtown at present which is 15km fromIt is over Lennoxtown at present which is 15km from
the airport. The angle of descent is 6the airport. The angle of descent is 6oo
..
What is the height of the plane ?What is the height of the plane ?
Example 2Example 2
o h
tan 6 =
15
o
h = 15× tan 6
h = 1.58km
SO
H CA
H TO
A
33. 20 Aug 201720 Aug 2017 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
www.mathsrevision.com
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
2.2. Use tan of an angle to solveUse tan of an angle to solve
find adjacent length.find adjacent length.
1. To use tan of the angle to
find adjacent length.
Angles & TrianglesAngles & Triangles
1.1. Write down tan ratio.Write down tan ratio.
34. www.mathsrevision.com
TrigonometryTrigonometryS3
Credit
20 Aug 201720 Aug 2017 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Use the tan ratio to calculate how far the ladder
is away from the building.
45o
12m
ladder
o opp 12
tan 45 = =
adj d
o
12
d =
tan 45
d = 12m
d m
SO
H CA
H TO
A
35. www.mathsrevision.com
TrigonometryTrigonometryS3
Credit
6o
20 Aug 201720 Aug 2017 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Aeroplane
a = 1.58 km
LennoxtownAirport
Q1. An aeroplane is preparing to land at Glasgow Airport. It is over
Lennoxtown at present. It is at a height of 1.58 km above the ground. It
‘s angle of descent is 6o
.
How far is it from the airport to Lennoxtown?
Example 2Example 2
o 1.58
tan 6 =
d
o
1.58
d =
tan 6
d = 15 km
SO
H CA
H TO
A
38. 20 Aug 201720 Aug 2017 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
www.mathsrevision.com
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
2.2. Use tan ratio to find an angle.Use tan ratio to find an angle.
1. To show how to find an
angle using tan ratio.
Angles & TrianglesAngles & Triangles
1.1. Write down tan ratio.Write down tan ratio.
39. www.mathsrevision.com
TrigonometryTrigonometryS3
Credit
20 Aug 201720 Aug 2017 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Use the tan ratio to calculate the angle that the
support wire makes with the ground.
xo
11m
o opp 11
tan x = =
adj 4
11
4
÷
o -1
x = tan
o o
x = 70
4 m
SO
H CA
H TO
A
40. www.mathsrevision.com
TrigonometryTrigonometryS3
Credit
20 Aug 201720 Aug 2017 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Use the tan ratio to find the angle of take-off.
xo
88m
o opp 88
tan x = =
adj 500
o
tan x = 0.176
o -1 o
x = tan (0.176) = 10
500 m
SO
H CA
H TO
A
43. 20 Aug 201720 Aug 2017 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
www.mathsrevision.com
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
2.2. Use sine ratio to find anUse sine ratio to find an
angle.angle.
1. Definite the sine ratio and
show how to find an angle
using this ratio.
Angles & TrianglesAngles & Triangles
1.1. Write down sine ratio.Write down sine ratio.
51. www.mathsrevision.comwww.mathsrevision.com
Starter QuestionsStarter Questions
www.mathsrevision.com
1 2 3
1. Explain why we can simply pick out Q
the 5 figure summary for the data
19, 15, 11, 22, 9, 12, 11
2. Show that the original price of a car is £9000
If it cos
, Q and Q
then find
ts £8100 after a discount of 10%
3. Alorry is travelling at 40mph.
It has travelled 60 miles.
How long has it taken to travel 60 miles.
S3
Credit
52. 20 Aug 201720 Aug 2017 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
www.mathsrevision.com
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
2.2. Use sine ratio to solveUse sine ratio to solve
REAL-LIFE problems.REAL-LIFE problems.
1. To show how to use the
sine ratio to solve
REAL-LIFE problems.
Angles & TrianglesAngles & Triangles
1.1. Write down sine ratio.Write down sine ratio.
53. www.mathsrevision.com
TrigonometryTrigonometryS3
Credit
SO
H CA
H TO
A
20 Aug 201720 Aug 2017 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
The support rope is 11.7m long. The angle between
the rope and ground is 70o
. Use the sine ratio to
calculate the height of the flag pole.
70o
h
o opp h
sin 70 = =
hyp 11.7
h × o
= 11.7 sin70
h = 11m
11.7m
54. www.mathsrevision.com
TrigonometryTrigonometryS3
Credit
SO
H CA
H TO
A
20 Aug 201720 Aug 2017 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Use the sine ratio to find the angle of the ramp.
xo
10m
o opp 10
sin x = =
hyp 20
o 10
sin x =
20
÷
o -1 o10
x = sin = 30
20
20 m
57. 20 Aug 201720 Aug 2017 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
www.mathsrevision.com
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
2.2. Use sine ratio to find theUse sine ratio to find the
hypotenuse.hypotenuse.
1. To show how to calculate
the hypotenuse using the
sine ratio.
Angles & TrianglesAngles & Triangles
1.1. Write down sine ratio.Write down sine ratio.
60. www.mathsrevision.comwww.mathsrevision.com
Starter QuestionsStarter Questions
www.mathsrevision.com
1 3
1. Explain why we can simply pick out Q
then find the 5 figure summary for the data
9, 5, 11, 2, 9, 2
2. Find the original price of a football
If it costs £20 after a discoun
and Q
t of 80%
3. Alorry is travelling at 50mph.
It has travelled 75 miles.
Show that the time taken is 1hr 30 mins.
S3
Credit
61. 20 Aug 201720 Aug 2017 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
www.mathsrevision.com
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
2.2. Use cosine ratio to find aUse cosine ratio to find a
length or angle.length or angle.
1. Definite the cosine ratio
and show how to find an
length or angle using this
ratio.
Angles & TrianglesAngles & Triangles
1.1. Write down cosine ratio.Write down cosine ratio.
68. www.mathsrevision.comwww.mathsrevision.com
The Three RatiosThe Three Ratios
Cosine
Sine
Tangent
Sine
Sine
Tangent
Cosine
Cosine
Sine
www.mathsrevision.com
S3
Credit
opposite
opposite opposite
adjacent
adjacent
adjacent
hypotenuse
hypotenuse
hypotenuse