What If I Know the Hypotenuse? Pg. 21 Sine and Cosine Ratios 5.6 What If I Know the Hypotenuse? Pg. 21 Sine and Cosine Ratios

5.6 – What If I Know the Hypotenuse? Sine and Cosine Ratios Today you are going to explore two more trigonometric ratios that involve the hypotenuse: Sine and Cosine.

Hypotenuse: Hypotenuse Opposite Side: Opposite Adjacent Side: Adjacent Side opposite right angle, longest side Hypotenuse Opposite Adjacent Side opposite slope angle (rise) Side touching slope angle, not the hypotenuse (run)

H O A sine tangent cosine opp hyp adj hyp opp adj sin = cos = tan = O Note: You cannot use trig ratios on the right angle

Find the ratio of the indicated angle. Write your answer as a fraction. 15 A 17 8 17 O 15 8

O 16 20 A H 12 20 16 12

Like the tangent, your calculator can give you both the sine and cosine ratios for any angle. Locate the "sin" and "cos" buttons on your calculator and use them to find the sine and cosine of 40°. Make sure you get the correct answers and are in degree mode. 0.64 0.77 0.84 sin 40°= _____ cos 40° = _______ tan 40° = _______

Label the sides with the reference angle Steps to solving with trig: 1. 2. 3. 4. Label the sides with the reference angle What side do you know? What side are you solving for? Identify which trig ratio to use

5.31 – TRIGONOMETRY For each triangle below, decide which side is opposite, adjacent, or the hypotenuse to the given angle. Then determine which of the three trig ratios will help you find x. Write and sole an equation. SOH-CAH-TOA might help.

H 1 O

H A 1

O A 1

O H 1

H O 1

H A 1

A H 1

5.32 – THE STREETS OF SAN FRANCISCO While traveling around the beautiful city of San Francisco, Julia climbed several steep streets. One of the steepest, Filbert Street, has a slope angle of 31.5° according to her guide book. Once Julia finished walking 100 feet up the hill, she decided to figure out how high she had climbed. Julia drew the diagram below to represent the situation.

a. Can the Pythagorean theorem be used to find the opposite and adjacent side? Why or why not? No, only know one side

b. Can special triangles be used to find the opposite and adjacent side? Why or why not? No, 31.5 isn’t special

Sine and cosine because we know the hypotenuse and one angle c. Can we use sine, cosine, or tangent to find the opposite and adjacent side? Why or why not? H O Sine and cosine because we know the hypotenuse and one angle A

d. Julia still wants to know how many feet she climbed vertically and horizontally when she walked up Filbert Street. Use one of your new trig ratios to find both parts of the missing triangle. H O 1 A

H O A 1

5.33 – EXACTLY! Martha arrived for her geometry test only to find that she forgot her calculator. She decided to complete as much of each problem as possible.

Set up equation multiplied divided a. In the first problem on the test, Martha was asked to find the length of x in the triangle shown at right. Using her algebra skills, she wrote and solved an equation. Her work is shown below. Explain what she did in each step. Set up equation multiplied divided

b. Martha's answer in part (a) is called an exact answer b. Martha's answer in part (a) is called an exact answer. Now use your calculator to help Martha find the approximate length of x. x = 68.62

c. In the next problem, Martha was asked to find y in the triangle at right. Find the exact answer for y without using a calculator. Then use a calculator to find an approximate value for y.

A O 1

Right Triangles Project Your Name Block# Pythagorean Theorem: Given 2 sides 45º– 45º– 90º 30º– 60º– 90º O H Sine – S A H Cosine – C O A Tangent – T sin-1, cos-1, tan-1

H O O H sin θ = O H Sine – S

H A A H cos θ = A H Cosine – C