How can you apply right triangle facts to solve real life problems? Trigonometry Essential Question How can you apply right triangle facts to solve real life problems?

Lesson 8-6 The Sine and Cosine Ratios (page 312) The sine ratio and cosine ratio relate the legs to the hypotenuse .

Review! hypotenuse leg leg The tangent ratio is the ratio of the lengths of the legs . B hypotenuse Review! c leg a A b C leg

opposite leg vs adjacent leg B In relationship to angle A … c opposite leg a A b C adjacent leg

opposite leg vs adjacent leg B In relationship to angle B … c adjacent leg a A b C opposite leg

Definition of Tangent Ratio B Review! c a A b C tangent of ∠A = tan A

Definition of Sine Ratio B c a A b C sine of ∠A = sin A

Definition of Cosine Ratio B c a A b C cosine of ∠A = cos A

B c a A b C

REMEMBER! B DO NOT FORGET! c a A b C SOH CAH TOA

Example 1 Express the sine and cosine of ∠A & ∠B as ratios. 13 5 ____ C A 12 A sin A = ______ (b) cos A = ______ (c) sin B = ______ (d) cos B = ______

Example 1 Express the sine and cosine of ∠A & ∠B as ratios. 13 A 5 ____ C A 12 O sin A = ______ (b) cos A = ______ (c) sin B = ______ (d) cos B = ______

Now try this with a calculator! Example 2 Use the trigonometry table, then use a calculator to find the approximate decimal values. “≈” means “is approximately equal to” 0.3746 Please note that these are only APPROXIMATIONS! (a) sin 22º ≈ __________ (b) cos 79º ≈ __________ 0.1908 Now try this with a calculator!

0.3746 0.1908 (a) sin 22º ≈ ____________ (b) cos 79º ≈ ____________ To enter this in your calculator you will need to use the SIN or COS function key. Enter SIN(22) then press ENTER (=) and round to 4 decimal places. 0.3746 (a) sin 22º ≈ ____________ (b) cos 79º ≈ ____________ Enter COS(79) then press ENTER (=) and round to 4 decimal places. 0.1908

Now try this with a calculator! … Example 2 Use the trigonometry table to find the approximate angle measures. “≈” means “is approximately equal to” 61º Please note that these are only APPROXIMATIONS! (c) sin ______ ≈ 0.8746 (d) cos _______ ≈ 0.7771 39º Now try this with a calculator!

61º 39º (c) sin ______ ≈ 0.8746 (d) cos _______ ≈ 0.7771 To enter this in your calculator you will need to use the inverse key or 2nd function key. Enter SIN-1(.8746) then press ENTER (=) and round to the nearest degree 61º (c) sin ______ ≈ 0.8746 (d) cos _______ ≈ 0.7771 Enter COS-1(.7771) then press ENTER (=) and round to the nearest degree 39º

What can you say about the values for the sine or cosine of an angle? The values for sine and cosine … Think about it … if the hypotenuse is the longest side and it is the denominator of the ratios, then it … … will always be less than 1. Enter SIN-1(1.5) then press ENTER (=) …

Example 3 (a) Find the value of x and y to the nearest integer Example 3 (a) Find the value of x and y to the nearest integer. x ≈ ______ y ≈ ______ 52 84 x 38º y

Example 3 (a) Find the value of x and y to the nearest integer Example 3 (a) Find the value of x and y to the nearest integer. x ≈ ______ y ≈ ______ 52 66 84 x 38º y

Example 3 (b) Find the value of x and y to the nearest integer Example 3 (b) Find the value of x and y to the nearest integer. x ≈ ______ y ≈ ______ 12 1 x x y 1 55º 7 7 14

Example 3 (b) Find the value of x and y to the nearest integer Example 3 (b) Find the value of x and y to the nearest integer. x ≈ ______ y ≈ ______ 12 10 x x y 55º 7 7 14

Example 4 (a): Find “n” to the nearest degree. n ≈ ______ 44 nº O 14 20 H

Example 4 (b) Find the measures of the 3 angles of a 3-4-5 ∆.

Example 4 (b) Find the measures of the 3 angles of a 3-4-5 ∆. ∴ a right triangle. continue

Example 4 (b) Find the measures of the 3 angles of a 3-4-5 ∆. 90º - 53º = 37º ∴ the angle measures are 37º, 53º, & 90º.

How can you apply right triangle facts to solve real life problems? Assignment Written Exercises on pages 314 to 316 RECOMMENDED: 1 to 9 odd numbers REQUIRED: 11 to 23 odd numbers Worksheet on Lessons 8-5 & 8-6 The Sine, Cosine, and Tangent Ratios How can you apply right triangle facts to solve real life problems?