Objective: Students will be able to… Use the sine, cosine, and tangent ratios to determine missing side lengths and angle measures in a right triangle. Right Triangle Trigonometry Sections 9.1 and 9.2

IN A RIGHT TRIANGLE…. There are ratios we can use to determine side lengths. These ratios are constant, no matter what the lengths for the sides of the triangle are. These ratios are called trigonometric ratios. Three of the trigonometric ratios are: Sine (sin) Cosine (cos) Tangent (tan)

TRIG RATIOS Hypotenuse B C A SIN= leg opposite of angle, hypotenuse COS= leg adjacent to angle, hypotenuse TAN= opposite leg, adjacent leg SOHCAHTOA

Write the trig ratios for the following: x z y B CA tan A =tan B= sin A = sin B = cos A =cos B = NOTICE—the sin A is the cos B, and the sin B is the cos A. For the 2 acute angles, the sin of one is the cos of the other.

Let’s put numbers in… Use the triangle to write each ratio. ALWAYS leave answers as a reduced ratio G 16 R T sin G = cos T= tan G = sin T= cos G= tan T= G R

Example Use the triangle to write each ratio. 60 Z Y X

Find the value of the trig ratio to the nearest ten- thousandth (4 decimal places) C B A 37

If given the angle measure, you can use a trig function to find a missing side length of a right triangle. x 25 M L Find x. K 57° Which trig ratio relates the given angle, and the 2 sides?? Set up equation: **Always keep at least 4 decimal places until you reach your final answer.

Examples: Find x ° 18 x x 30° 4

Example: To measure the height of a tree, Mrs. Shattuck walked 125 ft. from the tree, and measured a 32˚ angle from the ground to the top of the tree. Estimate the height of the tree.

Example: A 20 ft wire supporting a flagpole forms a 35˚ angle with the flagpole. To the nearest foot, how high is the flagpole?

Example: You are at the playground, and they just put in an awesome new slide. The slide is 25 ft long, and it creates a 57˚ angle with the ground. How high off the ground is the top of the slide?

If you need to find an angle in a right triangle given the side lengths, you use the inverse of the trig function: tan -1, sin -1, cos -1 tan -1 (.5) = x “The angle,x, whose tangent is.5” sin -1 (.7314)=x “The angle,x, whose sine is.7314” cos -1 (.5592)=x “The angle,x, whose cosine is.5592”

Fill in the blanks…. 1. cos __________ ≈ sin __________ ≈ tan___________ ≈.2309 In your calculator, enter cos -1 (.0175)

Find the indicated angle measures S T R A How would you now find the measure of angle T??

Find the measure of angle A. A 15 32

Example: A right triangle has a leg 1.5 units long and a hypotenuse 4.0 units long. Find the measures of its acute angles to the nearest degree.

Example: You are 200 ft from the base of a 150 ft building. What is the angle formed from the found where you are standing to the top of the building??