Lesson 12: Right Triangle Trigonometry

Lehman PTS-3 Summer 2020

1. Bridge to Calculus Workshop   Summer 2020 Lesson 12 Right Triangle Trigonometry “ A person who never made a mistake never tried anything new ." – Albert Einstein -

2. Right Triangle Definitions (1 of 2) Consider the right triangle given in the figure below, where the lengths of the sides are denoted by the positive numbers , and : How many ratios of different side lengths can we get?  

3. Right Triangle Definitions (2 of 2) The possible ratios are: And their reciprocals: A trigonometric ratio is a ratio of the lengths of two different sides of a right triangle. The tangent of an acute angle of a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle                 (i) (ii) (iii) (iv) (v) (vi)

4. The Tangent Ratio (1 of 2) Example 1. Answer the following questions based on right triangle below: Identify the leg opposite . Identify the leg adjacent to . Which angle of has a tangent of Solution.     a)   b)   c)  

5. The Tangent Ratio (2 of 2) Example 2. Find for each right triangle : Solution.   a)   b) c) a) b) c)    

6. Angle of Elevation (1 of 3) DEFINITION . An angle formed by a horizontal line and the line of sight to an object that is above the horizontal line is called the angle of elevation . The angle formed by the horizontal line and the line of sight to an object that is below the horizontal line is called the angle of depression .

7. Angle of Elevation (2 of 3) Example 3. A man stands meters from the trunk of a tree. The angle of elevation from eye level to the top of the tree is . The distance from the man’s eye level to the ground is meters. Given that .   a) Let be the distance from the man’s eye level to the top of the tree. Write an expression for the height of the tree in terms of .   b) Find to the nearest tenth of a meter   c) Find the height of the tree to the nearest tenth of a meter

8. Angle of Elevation (3 of 3) Solution. b) Express through :     It follows that:       a) Height of tree     c) Height of tree          

9. Tapeworm Head under Microscope

10. The Sine and Cosine Ratios (1 of 6) The sine of an acute angle of a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse :    

11. The Sine and Cosine Ratios (2 of 6) The cosine of an acute angle of a right triangle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse :    

12. The Sine and Cosine Ratios (3 of 6) Example 4. For triangle below, find the sine, cosine and tangent of : Solution.              

13. The Sine and Cosine Ratios (4 of 6) Example 5. For triangle below, find the value of to the nearest tenth, given that and : Solution.       Step 1 . Express and the known length of through a trigonometric ratio of . Which one?   Step 2 . Solve for :          

14. The Sine and Cosine Ratios (5 of 6) Example 6. You set a -foot ladder against a building. You want the angle between the ladder and the ground to be . To the nearest foot, how far from the building should you place the bottom of the ladder, g iven that , , and ? Solution.   Let be the distance from the base of the building to the bottom of the ladder.   Write an expression for the length of the ladder in terms of .  

15. The Sine and Cosine Ratios (6 of 6) Solution (cont’d). Write an expression for the length of the ladder in terms of .               It follows that:

16. The Pythagorean Theorem (1 of 3) Example 7. For triangle below, find the value : Solution. Triangle is called a - - triangle.   Step 1 . To determine , we need to know the length of side .   Step 2 . Evaluate :     Let the length of side be .   Then, by the Pythagorean theorem:       It follows that:        

17. The Pythagorean Theorem (2 of 3) Example 8. For triangle below, find the value : Solution. Triangle is called a - - triangle.   Step 1 . To determine , we need to know the length of side .   Step 2 . Evaluate :     Let the length of side be .   Then, by the Pythagorean theorem:     It follows that:           and    

18. The Pythagorean Theorem (3 of 3) Example 8. For the right triangle below, find the unknown length in simplest form: Solution.     Pythagoras’ theorem         Add   Take positive square roots of either side     Simplify       Find highest perfect square factor of   Split square roots Evaluate

19. Trigonometric Identities (1 of 2) Consider the right triangle below : We know that: Let us look at the ratio:       and that:                  

20. Trigonometric Identities (1 of 2) Write the Pythagorean Theorem for triangle below :       Pythagorean theorem Divide either side by           Split fraction on left-hand side     Property of exponents

21. Trigonometric Identities (2 of 2) From the previous slide, we had :     What trigonometric functions of are the ratios:     and       It follows that:       We write that as:       The above is called a Pythagorean trigonometric identity .