5_Trigonometry.pptx sin cos tan trig ratio

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1. Trigonometry

2. Warm Up Use the figure. 1. What kind of triangle is ? 2. Name the hypotenuse of . 3. Name the side opposite . 4. Name the side adjacent to that is not the hypotenuse. 5. Name the side adjacent to that is not the hypotenuse.  

3. Warm Up Use the figure. 1. What kind of triangle is ? 2. Name the hypotenuse of . 3. Name the side opposite . 4. Name the side adjacent to that is not the hypotenuse. 5. Name the side adjacent to that is not the hypotenuse.   right        

4. Standards for Mathematical Content G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.

5. MP5 Use appropriate tools strategically. MP7 Look for and make use of structure. Standards for Mathematical Practice

6. Lesson Goals ● Solve problems by using the trigonometric ratios for acute angles. ● Solve problems by using the inverse trigonometric ratio for acute angles.

7. Learn Trigonometry The word trigonometry comes from the Greek terms trigon , meaning triangle, and metron , meaning measure. So the study of trigonometry involves triangle measurement. A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. The names of the three most common trigonometric ratios are given on the next few slides.

8. Learn Trigonometry Key Concept: Trigonometric Ratios Sine: If is a right triangle, then the sine of each acute angle in is the ratio of the length of the leg opposite that angle to the length of the hypotenuse. or or Key Concept: Trigonometric Ratios

9. Learn Trigonometry Cosine: If is a right triangle, then the cosine of each acute angle in is the ratio of the length of the leg adjacent to that angle to the length of the hypotenuse. or or

10. Learn Trigonometry Tangent : If is a right triangle, then the tangent of each acute angle in is the ratio of the length of the leg opposite that angle to the length of the leg adjacent to that angle. or or

11. Example 1 Find Trigonometric Ratios Find sin   J , cos   J , tan   J , sin   K , cos   K , and tan   K . Express each ratio as a fraction and as a decimal to the nearest hundredth.

12. Example 1 Find Trigonometric Ratios

13. Example 1 Find Trigonometric Ratios Think About It! How are sin   J and cos   K related?

14. Example 2 Use a Special Right Triangle to Find Trigonometric Ratios Special right triangles can be used to find the sine, cosine, and tangent of , , and angles.  

15. Example 2 Use a Special Right Triangle to Find Trigonometric Ratios Use a special right triangle to express the sine of as a fraction and as a decimal to the nearest hundredth.  

16. Example 2 Use a Special Right Triangle to Find Trigonometric Ratios Using the Triangle Theorem, write the correct side lengths for each leg of the right triangle with as the length of the shorter leg.   Definition of sine ratio Substitution Use a calculator. Definition of sine ratio Substitution Use a calculator.

17. Example 3 Estimate Measures by Using Trigonometry ACCESSIBILITY Mathias builds a ramp so his sister can access the back door of their house. The -foot ramp to the house slopes upward from the ground at a angle. What is the horizontal distance between the foot of the ramp and the house? What is the height of the ramp?  

18. Example 3 Estimate Measures by Using Trigonometry Find the horizontal distance. Let The horizontal distance between the foot of the ramp and the house is , the measure of the leg adjacent to . The length of the ramp is the measure of the hypotenuse, feet. Because the lengths of the leg adjacent to a given angle and the hypotenuse are involved, write an equation using the cosine ratio.  

19. Example 3 Estimate Measures by Using Trigonometry Definition of cosine ratio Substitution Multiply each side by 12. Use a calculator. Definition of cosine ratio Substitution Multiply each side by 12. Use a calculator. The horizontal distance between the foot of the ramp and the house is about feet.  

20. Example 3 Estimate Measures by Using Trigonometry Find the height. The height of the ramp is , the measure of the leg opposite from . Because the lengths of the leg opposite to a given angle and the hypotenuse are involved, write an equation using a sine ratio.  

21. Example 3 Estimate Measures by Using Trigonometry Definition of sine ratio Substitution Multiply each side by 12. Use a calculator. Definition of sine ratio Substitution Multiply each side by 12. Use a calculator. The height y of the ramp is about feet or about inches.  

22. Learn Inverse Trigonometric Ratios If you know the value of a trigonometric ratio for an acute angle, you can use a calculator to find the measure of the angle, which is the inverse of the trigonometric ratio.

23. Learn Inverse Trigonometric Ratios Key Concept: Inverse Trigonometric Ratios Inverse Sine Inverse Cosine Inverse Tangent Words If is an acute angle and the sine of A is x , then the inverse sine of x is the measure of . If is an acute angle and the cosine of A is x , then the inverse cosine of x is the measure of . If is an acute angle and the tangent of A is x , then the inverse tangent of x is the measure of . Symbols If sin   A   =   x , then If cos   A   =   x , then . If tan   A   =   x , then . Key Concept: Inverse Trigonometric Ratios Inverse Sine Inverse Cosine Inverse Tangent Words Symbols

24. Example 4 Find Angle Measures by Using Inverse Trigonometric Ratios Use a calculator to find to the nearest tenth.  

25. Example 4 Find Angle Measures by Using Inverse Trigonometric Ratios The measures given are those of the leg adjacent to and the hypotenuse, so write an equation using the cosine ratio.   or Use the inverse cosine function. So, . Use a calculator. Use the inverse cosine function. Use a calculator.

26. Example 4 Find Angle Measures by Using Inverse Trigonometric Ratios Talk About It! What other method could you use to find ? Explain.  

27. Example 4 Find Angle Measures by Using Inverse Trigonometric Ratios Check Use a calculator to find to the nearest tenth.    

28. Example 4 Find Angle Measures by Using Inverse Trigonometric Ratios Check Use a calculator to find to the nearest tenth.    

29. Example 5 Solve a Right Triangle When you are given measurements to find the unknown angle and side measures of a triangle, this is known as solving a triangle . To solve a right triangle, you need to know: • two side lengths or • one side length and the measure of one acute angle.

30. Example 5 Solve a Right Triangle Solve the right triangle. Round side and angle measures to the nearest tenth.

31. Example 5 Solve a Right Triangle Use a sine ratio to find .   Use the inverse sine function. Use a calculator. Use the inverse sine function. Use a calculator. So, .  

32. Example 5 Solve a Right Triangle Use known angles to find .   Acute of rt are comp. Subtract from each side. So,  

33. Example 5 Solve a Right Triangle Use the Pythagorean Theorem to find RS . Pythagorean Theorem Substitution Simplify. Use a calculator. Pythagorean Theorem Substitution Simplify. Use a calculator. So, .  

34. Example 5 Solve a Right Triangle Check Solve the right triangle. Round side and angle measures to the nearest tenth. ; ;  

35. Example 5 Solve a Right Triangle Check Solve the right triangle. Round side and angle measures to the nearest tenth. ; ;  

36. Exit Ticket Can you solve a right triangle using the given information? Answer yes , no , or sometimes . If yes or sometimes , describe your method. 1. one side length 2. at least two side lengths 3. one acute angle measure and one side length 4. at least two angle measures

37. Exit Ticket Can you solve a right triangle using the given information? Answer yes , no , or sometimes . If yes or sometimes , describe your method. 1. one side length no 2. at least two side lengths yes; trigonometric ratios 3. one acute angle measure and one side length yes; trigonometric ratios or rules for special right triangles 4. at least two angle measures no