Lesson 8-6 The Sine and Cosine Ratios (page 312) The sine ratio and cosine ratio relate the legs to the hypotenuse. How can trigonometric ratios be used to find sides and angles of a triangle?

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

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

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

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

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

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

a A b B C c

a A b B C c REMEMBER! SOHCAHTOA DO NOT FORGET!

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

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

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

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

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

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 Enter COS -1 (.7771) then press ENTER (=) and round to the nearest degree (c)sin ______ ≈ (d)cos _______ ≈ º 39º

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

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

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

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

Example 3 (b) Find the value of x and y to the nearest integer. x ≈ ______ y ≈ ______ x 14 55º 12 y x 77 10

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

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

5 4 3 ∴ a right triangle. continue

5 4 3 Example 4 (b) Find the measures of the 3 angles of a ∆. nº ∴ the angle measures are 37º, 53º, & 90º. 90º - 53º = 37º

OPTIONAL Assignment Written Exercises on pages 314 to to 17 odd numbers ~ #20 is BONUS! ~ Trigonometry Worksheet #1 Lessons 8-5 & 8-6 The Sine, Cosine, and Tangent Ratios How can trigonometric ratios be used to find sides and angles of a triangle?