1. THE NATURE OF GEOMETRY Copyright © Cengage Learning. All rights reserved. 7

2. 7.5 Right-Triangle Trigonometry

3. 3 An important theorem from geometry, the Pythagorean theorem, has an important algebraic representation, and is important in our study of triangles.

4. 4 Right-Triangle Trigonometry A correctly labeled right triangle is shown in Figure 7.49. In a right triangle, the sides that are not the hypotenuse are sometimes called legs. Figure 7.49 Right triangle

5. 5 Example 1 – Build a right angle A carpenter wants to make sure that the corner of a closet is square (a right angle). If she measures out sides of 3 feet and 4 feet, how long should she make the diagonal (hypotenuse)? Solution: The length of the hypotenuse is the unknown, so use the Pythagorean theorem: The sides are 3 and 4.

6. 6 Example 1 – Solution She should make the diagonal 5 feet long. cont’d

7. 7 Trigonometric Ratios

8. 8 There are six possible ratios for the triangle shown in Figure 7.49. Figure 7.49 Right triangle

9. 9 Trigonometric Ratios The trigonometric ratios, are defined in the box.

10. 10 Example 2 – Find angles in a triangle using trigonometry Given a right triangle with sides of length 5 and 12, find the trigonometric ratios for the angles A and B. Show your answers in both common fraction and decimal fraction form, with decimals rounded to four places. Solution: First use the Pythagorean theorem to find the length of the hypotenuse.

11. 11 Example 2 – Solution sin A =  0.3846; cos A =  0.9231; tan A =  0.4167 sin B =  0.9231; cos B =  0.3846; tan B =  2.4 cont’d

12. 12 Example 3 – Exact values for 45° angle Find the cosine, sine, and tangent of 45°. Solution: If one of the angles of a right triangle is 45°, then the other acute angle must also be 45°(because the sum of the angles of a triangle is 180°).

13. 13 Example 3 – Solution Furthermore, since the base angles have the same measure, the triangle is isosceles. By the Pythagorean theorem, x 2 + x 2 = r 2 2x 2 = r 2 Now use the definition of the trigonometric ratios. cont’d Note that x is positive.

14. 14 Example 3 – Solution cont’d

15. 15 Inverse Trigonometric Ratios

16. 16 Inverse Trigonometric Ratios We can also use right-triangle trigonometry to find one of the acute angles if we know the trigonometric ratio. For example, suppose we know (as we do from Example 3) that tan  = 1 Also suppose that we do not know the angle . In other words, we ask, “What is the angle  ?” We answer by saying, “  is the angle whose tangent is 1.” In mathematics, we call this the inverse tangent and we write  = tan –1 1

17. 17 Inverse Trigonometric Ratios To find the angle , we turn to a calculator. Find the button labeled and press The display is 45, which means  = 45°.

18. 18 Inverse Trigonometric Ratios We now define the inverse trigonometric ratios for a right triangle.

19. 19 Example 6 – Find angles of a triangle using inverse trigonometric ratios Given a right triangle with sides of length 5 and 12, find the measures of the angles of this triangle. Solution: First use the Pythagorean theorem to find the length of the hypotenuse.

20. 20 Example 6 – Solution Also cont’d

21. 21 Example 6 – Solution Our task here is to find the measures of angles A and B. What is A? We might say, “A is the measure of the angle whose sine is ” This is the inverse sine, and we write cont’d

22. 22 Inverse Trigonometric Ratios The angle of elevation is the acute angle measured up from a horizontal line to the line of sight, whereas if we take the climber’s viewpoint, and measure from a horizontal down to the line of sight, we call this angle the angle of depression.

23. 23 Example 7 – Find the height of a tree from angle of elevation The angle of elevation to the top of a tree from a point on the ground 42 ft from its base is 33°. Find the height of the tree (to the nearest foot).

24. 24 Example 7 – Solution Let  = angle of elevation and h = height of tree. Then tan  = h = 42 tan 33°  27.28 The tree is 27 ft tall.