1. Geometry One is always a long way from solving a problem until one actually has the answer. Stephen Hawking Today: 9.5 Instruction Practice

2. Geometry - Yesterday One is always a long way from solving a problem until one actually has the answer. Stephen Hawking Assignment: Trig Notes

3. 9.5 Trigonometric Ratios Objectives: 1.Find the sine, cosine and tangent of acute angles. 2.Use trig ratios in real life. Vocabulary: Sine, Cosine, Tangent, Angle of Elevation

4. Finding Trig Ratios A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. The word trigonometry is derived from the ancient Greek language and means measurement of triangles. The three basic trigonometric ratios are sine, cosine, and tangent, which are abbreviated as sin, cos, and tan respectively.

5. Trigonometric Ratios Let ∆ABC be a right triangle. The since, the cosine, and the tangent of the acute angle A are defined as follows. sin A = Side opposite A hypotenuse = a c cos A = Side adjacent A hypotenuse = b c tan A = Side opposite A Side adjacent A = a b

6. Ex. 1: Finding Trig Ratios Compare the sine, the cosine, and the tangent ratios for A in each triangle beside. By the SSS Similarity Theorem, the triangles are similar. Their corresponding sides are in proportion which implies that the trigonometric ratios for A in each triangle are the same.

7. Ex. 1: Finding Trig Ratios LargeSmall sin A = opposite hypotenuse cosA = adjacent hypotenuse tanA = opposite adjacent 8 17 ≈ 0.4706 15 17 ≈ 0.8824 8 15 ≈ 0.5333 4 8.5 ≈ 0.4706 7.5 8.5 ≈ 0.8824 4 7.5 ≈ 0.5333 Trig ratios can be expressed as fractions or decimals.

8. Finding Trig Ratios - On Your OWN sin B = opposite hypotenuse cosB = adjacent hypotenuse tanB = opposite adjacent Take the big triangle from example #1, now focus on angle B. How does this change your labels?

9. Using a Calculator You can use a calculator to approximate the sine, cosine, and the tangent of 74. Make sure that your calculator is in degree mode. The table shows some sample keystroke sequences accepted by most calculators.

10. Sample keystrokes Sample keystroke sequences Sample calculator displayRounded Approximation 74 0.9612626950.9613 0.2756373550.2756 3.4874144443.4874 sin ENTER 74 74 COS ENTER 74 74 TAN ENTER

11. Using Trigonometric Ratios in Real-life Suppose you stand and look up at a point in the distance. Maybe you are looking up at the top of a tree as in Example 6. The angle that your line of sight makes with a line drawn horizontally is called angle of elevation.

12. Indirect Measurement  You are measuring the height of a Sitka spruce tree in Alaska. You stand 45 feet from the base of the tree. You measure the angle of elevation from a point on the ground to the top of the top of the tree to be 59°. To estimate the height of the tree, you can write a trigonometric ratio that involves the height h and the known length of 45 feet.

13. The math tan 59 ° = opposite adjacent tan 59 ° = h 45 45 tan 59° = h 45 (1.6643) ≈ h 74.9 ≈ h Write the ratio Substitute values Multiply each side by 45 Use a calculator or table to find tan 59 ° Simplify The tree is about 75 feet tall.

14. Estimating Distance Escalators. The escalator at the Wilshire/Vermont Metro Rail Station in Los Angeles rises 76 feet at a 30° angle. To find the distance d a person travels on the escalator stairs, you can write a trigonometric ratio that involves the hypotenuse and the known leg of 76 feet. 30 °

15. Now the math 30 ° sin 30 ° = opposite hypotenuse sin 30 ° = 76 d d sin 30° = 76 sin 30 ° 76 d = 0.5 76 d = d = 152 Write the ratio for sine of 30 ° Substitute values. Multiply each side by d. Divide each side by sin 30 ° Substitute 0.5 for sin 30 ° Simplify A person travels 152 feet on the escalator stairs.

16. Geometry One is always a long way from solving a problem until one actually has the answer. Stephen Hawking Assignment:  9.5 p 562 #9-41 Every Other Odd