Pre-5.1 - trigonometry ratios in right triangle and special right triangles.pptMariOsnolaSan
Right triangle trigonometry is based on ratios of the sides of right triangles. The six trigonometric functions - sine, cosine, tangent, cotangent, secant, and cosecant - are defined by ratios of the opposite, adjacent, and hypotenuse sides. To find a missing angle or side, the appropriate trigonometric ratio is selected based on the known sides and then calculated. There are special right triangle theorems for 45-45-90 and 30-60-90 triangles that allow determining side lengths based on one known side length.
Precalculus August 28 trigonometry ratios in right triangle and special righ...2236531
This document discusses right triangle trigonometry. It defines the six trigonometric functions - sine, cosine, tangent, cotangent, secant, and cosecant - which are ratios of the sides of a right triangle. The three sides are the hypotenuse, which is opposite the right angle, and the opposite and adjacent sides in relation to the acute angle. The document also provides examples of using trigonometric functions to find missing angles and sides of right triangles, including using special right triangle theorems for 45-45-90 and 30-60-90 triangles.
This document discusses right triangle trigonometry. It defines the six trigonometric functions as ratios of sides of a right triangle. The sides are the hypotenuse, adjacent side, and opposite side relative to an acute angle. It shows how to calculate trig functions for a given angle and how to find an unknown angle given two sides of a right triangle using inverse trig functions. Examples are provided to demonstrate solving for missing sides and angles of right triangles using trig ratios and the Pythagorean theorem.
Define the sine, cosine, and tangent ratios and their inverses
Find the measure of a side given a side and an angle
Find the measure of an angle given two sides
Use trig ratios to solve problems
This document provides an overview of trigonometric ratios and how to use them to solve problems involving right triangles. It defines the sine, cosine, and tangent ratios and provides the mnemonic device SOH CAH TOA to remember them. Examples are given of calculating trig ratios in right triangles and using them to solve application problems like finding the length of a ladder leaning against a wall.
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.
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 document provides information about trigonometric ratios and right triangles. It begins by defining a ratio as a comparison between two numbers. In trigonometry, ratios compare the sides of a right triangle. It then defines the three main trigonometric ratios - sine, cosine, and tangent - showing how each relates two sides of a right triangle. Additional information and examples are provided to explain calculating unknown angles and sides using trigonometric ratios and a scientific calculator.
Pre-5.1 - trigonometry ratios in right triangle and special right triangles.pptMariOsnolaSan
Right triangle trigonometry is based on ratios of the sides of right triangles. The six trigonometric functions - sine, cosine, tangent, cotangent, secant, and cosecant - are defined by ratios of the opposite, adjacent, and hypotenuse sides. To find a missing angle or side, the appropriate trigonometric ratio is selected based on the known sides and then calculated. There are special right triangle theorems for 45-45-90 and 30-60-90 triangles that allow determining side lengths based on one known side length.
Precalculus August 28 trigonometry ratios in right triangle and special righ...2236531
This document discusses right triangle trigonometry. It defines the six trigonometric functions - sine, cosine, tangent, cotangent, secant, and cosecant - which are ratios of the sides of a right triangle. The three sides are the hypotenuse, which is opposite the right angle, and the opposite and adjacent sides in relation to the acute angle. The document also provides examples of using trigonometric functions to find missing angles and sides of right triangles, including using special right triangle theorems for 45-45-90 and 30-60-90 triangles.
This document discusses right triangle trigonometry. It defines the six trigonometric functions as ratios of sides of a right triangle. The sides are the hypotenuse, adjacent side, and opposite side relative to an acute angle. It shows how to calculate trig functions for a given angle and how to find an unknown angle given two sides of a right triangle using inverse trig functions. Examples are provided to demonstrate solving for missing sides and angles of right triangles using trig ratios and the Pythagorean theorem.
Define the sine, cosine, and tangent ratios and their inverses
Find the measure of a side given a side and an angle
Find the measure of an angle given two sides
Use trig ratios to solve problems
This document provides an overview of trigonometric ratios and how to use them to solve problems involving right triangles. It defines the sine, cosine, and tangent ratios and provides the mnemonic device SOH CAH TOA to remember them. Examples are given of calculating trig ratios in right triangles and using them to solve application problems like finding the length of a ladder leaning against a wall.
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.
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 document provides information about trigonometric ratios and right triangles. It begins by defining a ratio as a comparison between two numbers. In trigonometry, ratios compare the sides of a right triangle. It then defines the three main trigonometric ratios - sine, cosine, and tangent - showing how each relates two sides of a right triangle. Additional information and examples are provided to explain calculating unknown angles and sides using trigonometric ratios and a scientific calculator.
This document summarizes trigonometry concepts including defining trig ratios (sine, cosine, tangent) for right triangles and their inverses. It explains how to use trig ratios to find missing sides or angles of right triangles by setting up equations. Examples are provided to calculate trig ratios in a triangle, find side lengths using trig ratios, and find a missing angle. Mnemonics like SOH-CAH-TOA are suggested to memorize the trig ratio relationships.
Three-Dimensional Geometry discusses spatial relations and three-dimensional figures. It explains that three-dimensional figures have faces, edges, and vertices. The document provides examples and formulas for calculating the volumes of prisms, cylinders, cones, pyramids and cubes. It also discusses surface area and provides examples and formulas for calculating surface areas of prisms and cylinders.
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 the Pythagorean theorem and right triangles:
1) It defines the key terms related to right triangles - the legs are the two sides adjacent to the right angle, and the hypotenuse is the side opposite the right angle.
2) It presents the Pythagorean theorem formula - a2 + b2 = c2, where a and b are the lengths of the legs and c is the length of the hypotenuse.
3) It provides examples of using the theorem to determine the length of missing sides of right triangles.
This document discusses trigonometric ratios and solving trigonometric equations. It provides the definitions of sine, cosine, and tangent ratios relating the opposite, adjacent, and hypotenuse sides of a right triangle. An example problem is worked through to solve for the unknown side of a triangle using the sine ratio, substituting known values and rearranging the equation by multiplying both sides by the variable. A second example is presented using the tangent ratio to solve another trigonometric equation for the unknown side.
This document discusses trigonometric ratios and solving trigonometric equations. It provides the definitions of sine, cosine, and tangent ratios relating the opposite, adjacent, and hypotenuse sides of a right triangle. An example problem is worked through to solve for the unknown side of a triangle using the sine ratio, substituting known values and rearranging the equation by multiplying both sides by the variable. A second example is presented using the tangent ratio to solve another trigonometric equation for the unknown side.
This document provides information on trigonometry, including the sine rule, cosine rule, and basic trigonometry concepts for right triangles. It defines sine, cosine, and tangent as ratios of sides of a right triangle to the hypotenuse. The sine and cosine rules allow calculation of unknown sides or angles given certain known values. The cosine rule has two forms, one for finding a side and one for finding an angle. Examples are provided to demonstrate application of these trigonometric rules and concepts to solve for unknown values in right triangles.
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 presents information about various geometric shapes and formulas for calculating dimensions such as perimeter, area, surface area, and volume. It defines length, width, and height. Formulas are provided for calculating the perimeter and area of rectangles, squares, triangles, parallelograms, trapezoids, and circles. Surface area and volume formulas are given for cubes, cuboids, cylinders, cones, spheres, and other three-dimensional shapes.
Trigonometry deals with relationships between sides and angles in right triangles. The three main trigonometric functions are sine, cosine, and tangent, which provide ratios of sides of a right triangle based on a given angle. A scientific calculator can be used to find trigonometric ratios for any angle, as well as to calculate missing angles or sides of a right triangle when two pieces of information are known.
The document provides information on trigonometric ratios and how to use them to solve for missing angles and sides of right triangles. It defines the sine, cosine, and tangent ratios using opposite, adjacent, and hypotenuse side lengths relative to a designated angle θ. It gives examples of setting up and evaluating trig ratios, finding missing sides and angles of triangles using inverse trig functions, and solving application problems involving angles of elevation/depression.
The document discusses different types of three-dimensional shapes studied in solid geometry. It provides definitions and examples of cubes, rectangular prisms, cylinders, spheres, cones, and pyramids. It also gives the formulas for calculating the volume and surface area of these shapes. For each shape, it provides examples of applying the formulas to solve volume and surface area problems.
This document provides instruction on calculating the surface areas of prisms and cylinders. It defines key terms like lateral face, altitude, and axis. It presents formulas for calculating the surface area of right rectangular prisms and right cylinders. Examples are provided to demonstrate calculating surface areas of various prisms and cylinders. The document also addresses how surface areas change with modifications to dimensions and provides a word problem comparing the surface areas of different shapes.
The document provides information about the Pythagorean theorem and its applications. It defines the Pythagorean theorem as the square of the hypotenuse of a right triangle being equal to the sum of the squares of the other two sides. It gives examples of Pythagorean triples and how to use the theorem to solve for missing sides of right triangles. It also discusses classifying triangles as right, obtuse, or acute using the theorem and covers special right triangles.
This document discusses formulas and methods for finding the areas and perimeters of parallelograms, triangles, trapezoids, rhombuses, and kites. It provides examples of using these formulas to calculate side lengths, heights, areas, and perimeters when given certain measurements of the shapes. It also includes practice problems for students to solve.
The document provides instructions on using trigonometric functions to find missing angles or sides of right triangles. It begins with reminders of trigonometric definitions and ratios. Examples are then given to demonstrate finding a missing side using the Pythagorean theorem or a trigonometric ratio, and finding a missing angle using an inverse trigonometric function. Tips are provided on determining which method to use based on the information given in a problem.
The document provides information on surface area and volume formulas and calculations for basic 3D shapes including prisms, cubes, cylinders, cones, and spheres. It defines key terms like surface area and volume and provides example calculations and formulas for finding the surface area and volume of cubes, rectangular prisms, cylinders, cones, and spheres. Diagrams and examples are included to illustrate the different shapes and how to set up the surface area and volume calculations.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Training: ISO/IEC 27001 Information Security Management System - EN | PECB
ISO/IEC 42001 Artificial Intelligence Management System - EN | PECB
General Data Protection Regulation (GDPR) - Training Courses - EN | PECB
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This document summarizes trigonometry concepts including defining trig ratios (sine, cosine, tangent) for right triangles and their inverses. It explains how to use trig ratios to find missing sides or angles of right triangles by setting up equations. Examples are provided to calculate trig ratios in a triangle, find side lengths using trig ratios, and find a missing angle. Mnemonics like SOH-CAH-TOA are suggested to memorize the trig ratio relationships.
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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.
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1) It defines the key terms related to right triangles - the legs are the two sides adjacent to the right angle, and the hypotenuse is the side opposite the right angle.
2) It presents the Pythagorean theorem formula - a2 + b2 = c2, where a and b are the lengths of the legs and c is the length of the hypotenuse.
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This document discusses trigonometric ratios and solving trigonometric equations. It provides the definitions of sine, cosine, and tangent ratios relating the opposite, adjacent, and hypotenuse sides of a right triangle. An example problem is worked through to solve for the unknown side of a triangle using the sine ratio, substituting known values and rearranging the equation by multiplying both sides by the variable. A second example is presented using the tangent ratio to solve another trigonometric equation for the unknown side.
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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.
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Trigonometry deals with relationships between sides and angles in right triangles. The three main trigonometric functions are sine, cosine, and tangent, which provide ratios of sides of a right triangle based on a given angle. A scientific calculator can be used to find trigonometric ratios for any angle, as well as to calculate missing angles or sides of a right triangle when two pieces of information are known.
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Find out more about ISO training and certification services
Training: ISO/IEC 27001 Information Security Management System - EN | PECB
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Webinars: https://pecb.com/webinars
Article: https://pecb.com/article
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For more information about PECB:
Website: https://pecb.com/
LinkedIn: https://www.linkedin.com/company/pecb/
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Slideshare: http://www.slideshare.net/PECBCERTIFICATION
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2. The longest side in a right-angled
triangle is called the hypotenuse (hyp).
xo
The side next to the angle is
called the adjacent (adj).
The side opposite
the angle
is called the
opposite (opp).
5. 24o
x
6 cm
Example
1 Find the value of x.
opp
adj
tan24o
opp
adj
tan24o
x
6
x 6 tan24o
x 2.67 cm (to 3 s.f.)
replace opp by x and adj by 6
multiply both sides by 6
6. 73o
x
8 cm
Example
2 Find the value of x.
adj
hyp
cos73o
adj
hyp
cos73o
x
8
x 8 cos73o
x 2.34 cm (to 3 s.f.)
replace adj by x and hyp by 8
multiply both sides by 8
7. 40o
x 5 cm
Example
3 Find the value of x.
opp
hyp
sin40o
opp
hyp
sin40o
x
5
x 5 sin40o
x 3.21 cm (to 3 s.f.)
replace opp by x and hyp by 5
multiply both sides by 5
8. 65o
x
15 cm
Example
4 Find the value of x.
adj
hyp
cos65o
adj
hyp
cos65o
15
x
x cos65o
15
x 35.5 cm (to 3 s.f.)
replace adj by 15 and hyp by x
multiply both sides by x
x
15
cos65o
divide both sides by cos 65o
10. xo
5 cm 9 cm
Example
1 Find the value of x.
opp
hyp
sinx
opp
hyp
sinx
5
9
1 5
sin
9
x
x 33.7o
(to 3 s.f.)
replace opp by 5 and hyp by 9
to find x use the sin-1 button on your calculator
11. xo
5 cm
8 cm
Example
2 Find the value of x.
adj
hyp
cos x
adj
hyp
cos x
5
8
1 5
cos
8
x
x 51.3o
(to 3 s.f.)
replace adj by 5 and hyp by 8
to find x use the cos-1 button on your calculator
12. xo
3 cm
6 cm
Example
3 Find the value of x.
opp
adj
tanx
opp
adj
tanx
3
6
1 3
tan
6
x
x 26.6o
(to 3 s.f.)
replace opp by 3 and adj by 6
to find x use the tan-1 button on your calculator
14. To find the angle
between AG and the
base you need to
look at triangle AGC.
A B
7 cm
C
E
G
H
D
F
6 cm
5 cm
A B
7 cm
C
E
G
H
D
F
6 cm
5 cm
First you need to
calculate AC using
Pythagoras on
triangle ABC.
AC2
72
62
AC2
85
85
AC
85
Now use
trigonometry on
triangle ACG.
5
tan
85
x
1 5
tan
85
x
x 28.5o
(to 3 s.f.)
A C
G
5 cm
x