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3f angles
This document discusses different types of angles: right angles measure 90 degrees, acute angles are less than 90 degrees, obtuse angles are greater than 90 degrees, and straight angles follow a straight line. It provides vocabulary on how angles are named and composed of two rays with a common vertex. The document directs the reader to classwork on identifying angle types and asks where angles are seen in everyday life.
Pythagorean triples presentation
This document discusses Pythagorean triples, which are sets of three positive integers a, b, and c that satisfy the Pythagorean theorem a2 + b2 = c2. It provides examples of Pythagorean triples like (3, 4, 5) and explains Euclid's proof that there are infinitely many such triples. The document also describes properties of Pythagorean triples and how to construct them using formulas involving positive integers m and n. Finally, it mentions that the list provided only includes the first or "primitive" Pythagorean triple for each unique combination and not their multiples.
8.2 classifying triangles 1
The document provides information about classifying triangles based on their angles and sides. It defines different types of triangles such as acute, right, obtuse, equilateral, isosceles, and scalene triangles. It explains that all triangles have a sum of 180 degrees for their interior angles and can be used to find a missing third angle if two angles are given. Examples are provided to demonstrate classifying triangles and determining if a set of angle measures could define a triangle.
5.1.3 angles and their measure
This document contains formulas for arc length, area of a sector, linear speed, and angular speed. It provides examples of using these formulas to calculate the linear speed and angular speed of a Ferris wheel and test road speeds using a spin balancer on car wheels. Students are assigned problems from the textbook involving these concepts.
3D TRIG RATIOS
IGCSE F5 3D TRIGONOMETRIC RATIOS SAMPLE PROBLEMS AND SOLUTIONS.
THESE HAVE BEEN DONE BASED ON THE PYTHAGORAS THEOREM. COSINE RULE MAY HAVE BEEN USED BUT IT TARGETS THE LAY STUDENT.
Pythagorean triples
Pythagoras’s 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 document provides examples of Pythagorean triples, which are sets of three integers that satisfy the Pythagorean theorem, such as 3, 4, 5. It also lists several other common Pythagorean triples and provides a method for calculating additional triples using basic algebra. Finally, the document includes one person's opinion that while Pythagorean's theorem can be useful in situations like artillery firing, they rarely find it useful in everyday life.
Wordproblem
The document discusses trigonometric ratios and solving word problems involving angles of elevation and depression. It provides examples of calculating heights or distances using trigonometric ratios like tangent given angles of elevation or depression and known lengths of the adjacent or opposite sides of right triangles. Specifically, it gives the example of calculating the height of a tree as 8.49 meters using the tangent of the angle of elevation of 23 degrees from a point 20 meters from the base of the tree.
Trigonometry: Solving Triangles
The document summarizes two laws for solving triangles - the Law of Cosines and the Law of Sines. It also discusses an ambiguous case of the Law of Sines. The Law of Cosines can be used to find a missing side given two sides and the included angle, or to find a missing angle given all three sides. The Law of Sines can be used to find a missing side or angle given two angles and the side opposite one of them. The ambiguous case allows finding a missing angle given two sides and the angle opposite one of them.
Recommended
3f angles
This document discusses different types of angles: right angles measure 90 degrees, acute angles are less than 90 degrees, obtuse angles are greater than 90 degrees, and straight angles follow a straight line. It provides vocabulary on how angles are named and composed of two rays with a common vertex. The document directs the reader to classwork on identifying angle types and asks where angles are seen in everyday life.
Pythagorean triples presentation
This document discusses Pythagorean triples, which are sets of three positive integers a, b, and c that satisfy the Pythagorean theorem a2 + b2 = c2. It provides examples of Pythagorean triples like (3, 4, 5) and explains Euclid's proof that there are infinitely many such triples. The document also describes properties of Pythagorean triples and how to construct them using formulas involving positive integers m and n. Finally, it mentions that the list provided only includes the first or "primitive" Pythagorean triple for each unique combination and not their multiples.
8.2 classifying triangles 1
The document provides information about classifying triangles based on their angles and sides. It defines different types of triangles such as acute, right, obtuse, equilateral, isosceles, and scalene triangles. It explains that all triangles have a sum of 180 degrees for their interior angles and can be used to find a missing third angle if two angles are given. Examples are provided to demonstrate classifying triangles and determining if a set of angle measures could define a triangle.
5.1.3 angles and their measure
This document contains formulas for arc length, area of a sector, linear speed, and angular speed. It provides examples of using these formulas to calculate the linear speed and angular speed of a Ferris wheel and test road speeds using a spin balancer on car wheels. Students are assigned problems from the textbook involving these concepts.
3D TRIG RATIOS
IGCSE F5 3D TRIGONOMETRIC RATIOS SAMPLE PROBLEMS AND SOLUTIONS.
THESE HAVE BEEN DONE BASED ON THE PYTHAGORAS THEOREM. COSINE RULE MAY HAVE BEEN USED BUT IT TARGETS THE LAY STUDENT.
Pythagorean triples
Pythagoras’s 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 document provides examples of Pythagorean triples, which are sets of three integers that satisfy the Pythagorean theorem, such as 3, 4, 5. It also lists several other common Pythagorean triples and provides a method for calculating additional triples using basic algebra. Finally, the document includes one person's opinion that while Pythagorean's theorem can be useful in situations like artillery firing, they rarely find it useful in everyday life.
Wordproblem
The document discusses trigonometric ratios and solving word problems involving angles of elevation and depression. It provides examples of calculating heights or distances using trigonometric ratios like tangent given angles of elevation or depression and known lengths of the adjacent or opposite sides of right triangles. Specifically, it gives the example of calculating the height of a tree as 8.49 meters using the tangent of the angle of elevation of 23 degrees from a point 20 meters from the base of the tree.
Trigonometry: Solving Triangles
The document summarizes two laws for solving triangles - the Law of Cosines and the Law of Sines. It also discusses an ambiguous case of the Law of Sines. The Law of Cosines can be used to find a missing side given two sides and the included angle, or to find a missing angle given all three sides. The Law of Sines can be used to find a missing side or angle given two angles and the side opposite one of them. The ambiguous case allows finding a missing angle given two sides and the angle opposite one of them.
Pythagorean triples
The document discusses Pythagorean triples and the Pythagorean theorem. It provides:
1) A brief history of the Pythagorean theorem, which was discovered by Pythagoras and is also known to have been used in ancient India and Maya civilizations.
2) An explanation that Pythagorean triples are sets of integers that satisfy the Pythagorean theorem relationship a2 + b2 = c2.
3) Examples of the 3-4-5 triangle as the simplest Pythagorean triple and the non-Pythagorean triple of 3-7-9.
Solving word problems ~ complementary and supplementary angles
The document contains examples of solving word problems involving complementary angles, supplementary angles, and mixed problems involving both. Several word problems are presented and solved involving an angle (x), its complement (90-x), and supplement (180-x). Additional word problems at the end involve finding the measure of an angle given a relationship to its complement or supplement.
Angles
This document discusses angle measurements using degrees and different types of angles. It defines acute angles as between 0 and 90 degrees, obtuse angles as between 90 and 180 degrees, and right angles as 90 degrees. It also defines complementary angles as having a sum of 90 degrees, supplementary angles as having a sum of 180 degrees, and vertical angles as having the same degree measurement. Examples are provided to illustrate each angle type.
Complementary And Supplementary Angles
Complementary angles add up to 90 degrees, while supplementary angles add up to 180 degrees. The document provides examples of finding complementary angles by subtracting an acute angle from 90 degrees, and finding supplementary angles by subtracting an acute angle from 180 degrees. It concludes by thanking the audience.
Pythagorean triples
Pythagorean triples are sets of three whole numbers that satisfy the Pythagorean theorem. A primitive Pythagorean triple has no common factors between the three numbers. Euclid developed a formula to generate primitive Pythagorean triples using two integers where one is odd and they are relatively prime. The document discusses using Euclid's formula to find primitive Pythagorean triples and prove the formula, as well as properties of primitive Pythagorean triples.
Law of tangent
The document discusses the law of tangents, which sets up a ratio relationship between the sum and difference of two sides of a triangle and the tangents of half the sum and differences of the angles opposite the sides. An example problem demonstrates using the law of tangents to solve for a missing side of a triangle given two sides and one angle measure. The full working of the problem is shown, applying the tangent law formulas and then using the law of sines to find the final missing side measurement.
angles of parallel line cut by a transversal line
This document discusses angles formed when parallel lines are cut by a transversal line. It defines key terms like parallel lines, transversal, and describes angle pairs that are formed - alternate interior angles, alternate exterior angles, corresponding angles, interior angles on the same side of the transversal, and vertical angles. It notes that alternate interior angles, alternate exterior angles, and corresponding angles are congruent, while interior angles on the same side of the transversal are supplementary. Vertical angles are always equal.
S3 3 trigonometry USES
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.
Chapter9 trignometry
This document provides information about trigonometric ratios and their relationships to angles in the unit circle. It defines the trigonometric ratios of sine, cosine, and tangent. It discusses the four quadrants of the unit circle and identifies the signs of the trig ratios in each quadrant. Several examples are provided to illustrate calculating trig ratios based on the quadrant an angle falls in. The document also discusses graphs of the sine, cosine, and tangent functions over one period from 0 to 360 degrees. It provides exercises involving identifying quadrants, calculating trig ratios, relating angles in different quadrants, and solving trigonometric equations. Finally, it includes several past SPM exam questions involving applying trigonometric concepts in geometric contexts.
14 4 secant-tangent angles
Secant-tangent angles are formed when a secant line intersects a tangent line either on or outside of a circle. If the vertex is outside the circle, the degree measure is half the difference between the intercepted arcs. If the vertex is on the circle, the degree measure is half the intercepted arc. When two tangent lines form an angle, the degree measure is also half the difference between the intercepted arcs.
Parallel Lines & Related Angles(1 Na)
This document summarizes key concepts about angles and parallel lines from Chapter 7 including:
- Identifying different types of angles such as acute, obtuse, right, reflex
- Properties of angles including complementary, supplementary, adjacent, vertical
- Properties of angles formed by parallel lines and transversals including corresponding angles, alternate angles, interior angles
- Examples are provided to illustrate different types of angles and their properties
- Homework assignments are listed asking students to practice finding unknown angles using properties of angles and parallel lines.
Measure of an angles
The document defines different types of angles and how to measure them using a protractor. It explains that angles can be measured in degrees from 0° to 360° and defines right angles as being 90°, acute angles as less than 90°, and obtuse angles as greater than 90° but less than 180°. A straight angle is 180°. Examples are given measuring various angles and identifying their type.
Trigonometry
The document discusses trigonometry and its key concepts. It defines trigonometry as the study of relationships between sides and angles of triangles. It then covers right triangles, trigonometric ratios, the Pythagorean theorem, and ratios for common angles like 45, 60, and 90 degrees. Key formulas discussed include sine, cosine, tangent, cotangent, secant, and cosecant. Relationships between the ratios are also summarized.
Parallel Lines with Transversals
The document discusses parallel lines and transversals. It defines parallel lines and introduces the symbol || to represent parallel lines. It defines a transversal as a line that intersects two or more other lines. It identifies and defines exterior angles, interior angles, alternate interior angles, alternate exterior angles, and corresponding angles that are formed when a transversal intersects two lines. It provides examples of determining whether statements about angle relationships formed by lines and transversals are true or false.
Trig 1 lesson 4 tangent
Defining the Tangent ratio in Trigonometry.
Small changes between slides so can be clicked through quickly.
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Pythagorean triples
The document discusses Pythagorean triples and the Pythagorean theorem. It provides:
1) A brief history of the Pythagorean theorem, which was discovered by Pythagoras and is also known to have been used in ancient India and Maya civilizations.
2) An explanation that Pythagorean triples are sets of integers that satisfy the Pythagorean theorem relationship a2 + b2 = c2.
3) Examples of the 3-4-5 triangle as the simplest Pythagorean triple and the non-Pythagorean triple of 3-7-9.
Solving word problems ~ complementary and supplementary angles
The document contains examples of solving word problems involving complementary angles, supplementary angles, and mixed problems involving both. Several word problems are presented and solved involving an angle (x), its complement (90-x), and supplement (180-x). Additional word problems at the end involve finding the measure of an angle given a relationship to its complement or supplement.
Angles
This document discusses angle measurements using degrees and different types of angles. It defines acute angles as between 0 and 90 degrees, obtuse angles as between 90 and 180 degrees, and right angles as 90 degrees. It also defines complementary angles as having a sum of 90 degrees, supplementary angles as having a sum of 180 degrees, and vertical angles as having the same degree measurement. Examples are provided to illustrate each angle type.
Complementary And Supplementary Angles
Complementary angles add up to 90 degrees, while supplementary angles add up to 180 degrees. The document provides examples of finding complementary angles by subtracting an acute angle from 90 degrees, and finding supplementary angles by subtracting an acute angle from 180 degrees. It concludes by thanking the audience.
Pythagorean triples
Pythagorean triples are sets of three whole numbers that satisfy the Pythagorean theorem. A primitive Pythagorean triple has no common factors between the three numbers. Euclid developed a formula to generate primitive Pythagorean triples using two integers where one is odd and they are relatively prime. The document discusses using Euclid's formula to find primitive Pythagorean triples and prove the formula, as well as properties of primitive Pythagorean triples.
Law of tangent
The document discusses the law of tangents, which sets up a ratio relationship between the sum and difference of two sides of a triangle and the tangents of half the sum and differences of the angles opposite the sides. An example problem demonstrates using the law of tangents to solve for a missing side of a triangle given two sides and one angle measure. The full working of the problem is shown, applying the tangent law formulas and then using the law of sines to find the final missing side measurement.
angles of parallel line cut by a transversal line
This document discusses angles formed when parallel lines are cut by a transversal line. It defines key terms like parallel lines, transversal, and describes angle pairs that are formed - alternate interior angles, alternate exterior angles, corresponding angles, interior angles on the same side of the transversal, and vertical angles. It notes that alternate interior angles, alternate exterior angles, and corresponding angles are congruent, while interior angles on the same side of the transversal are supplementary. Vertical angles are always equal.
S3 3 trigonometry USES
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.
Chapter9 trignometry
This document provides information about trigonometric ratios and their relationships to angles in the unit circle. It defines the trigonometric ratios of sine, cosine, and tangent. It discusses the four quadrants of the unit circle and identifies the signs of the trig ratios in each quadrant. Several examples are provided to illustrate calculating trig ratios based on the quadrant an angle falls in. The document also discusses graphs of the sine, cosine, and tangent functions over one period from 0 to 360 degrees. It provides exercises involving identifying quadrants, calculating trig ratios, relating angles in different quadrants, and solving trigonometric equations. Finally, it includes several past SPM exam questions involving applying trigonometric concepts in geometric contexts.
14 4 secant-tangent angles
Secant-tangent angles are formed when a secant line intersects a tangent line either on or outside of a circle. If the vertex is outside the circle, the degree measure is half the difference between the intercepted arcs. If the vertex is on the circle, the degree measure is half the intercepted arc. When two tangent lines form an angle, the degree measure is also half the difference between the intercepted arcs.
Parallel Lines & Related Angles(1 Na)
This document summarizes key concepts about angles and parallel lines from Chapter 7 including:
- Identifying different types of angles such as acute, obtuse, right, reflex
- Properties of angles including complementary, supplementary, adjacent, vertical
- Properties of angles formed by parallel lines and transversals including corresponding angles, alternate angles, interior angles
- Examples are provided to illustrate different types of angles and their properties
- Homework assignments are listed asking students to practice finding unknown angles using properties of angles and parallel lines.
Measure of an angles
The document defines different types of angles and how to measure them using a protractor. It explains that angles can be measured in degrees from 0° to 360° and defines right angles as being 90°, acute angles as less than 90°, and obtuse angles as greater than 90° but less than 180°. A straight angle is 180°. Examples are given measuring various angles and identifying their type.
Trigonometry
The document discusses trigonometry and its key concepts. It defines trigonometry as the study of relationships between sides and angles of triangles. It then covers right triangles, trigonometric ratios, the Pythagorean theorem, and ratios for common angles like 45, 60, and 90 degrees. Key formulas discussed include sine, cosine, tangent, cotangent, secant, and cosecant. Relationships between the ratios are also summarized.
Parallel Lines with Transversals
The document discusses parallel lines and transversals. It defines parallel lines and introduces the symbol || to represent parallel lines. It defines a transversal as a line that intersects two or more other lines. It identifies and defines exterior angles, interior angles, alternate interior angles, alternate exterior angles, and corresponding angles that are formed when a transversal intersects two lines. It provides examples of determining whether statements about angle relationships formed by lines and transversals are true or false.
Trig 1 lesson 4 tangent
Defining the Tangent ratio in Trigonometry.
Small changes between slides so can be clicked through quickly.
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Solving word problems ~ complementary and supplementary angles
Solving word problems ~ complementary and supplementary angles