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Objective: To use the sine, cosine, and tangent ratios to determine missing side lengths in a right triangle. Right Triangle Trigonometry Sections 9.1 and 9.2

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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)

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

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Write the trig ratios for the following: x z y B CA tan A =tan B= sin A = sin B = cos A =cos B =

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Let’s put numbers in… Use the triangle to write each ratio. 12 20 G 16 RT sin G = cos T= tan G = sin T= cos G= tan T=

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Example Use the triangle to write each ratio. 60 Z 87 63 Y X

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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:

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Examples: Find x. 1.2. 58° 18 x x 30° 4

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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.

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

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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?

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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”

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Fill in the blanks…. 1. cos __________ ≈.0175 2. sin __________ ≈.9659 3. tan___________ ≈.2309 In your calculator, enter cos -1 (.0175)

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Find the indicated angle measures. 1. 2. S 41 47 T R 17 41 A How would you now find the measure of angle T??

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Find the measure of angle A. A 15 32

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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.

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

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Angles of Elevation and Depression Angle of Elevation-- the angle that an observer would raise his or her line of sight above a horizontal line in order to see an object. Angle of Depression-- If an observer were above and needed to look down, the angle of depression would be the angle that the person would need to lower his or her line of sight. *Why are the angles of elevation and depression between the same two objects congruent?*

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Describe each angle as it relates to the situation 1 2 3 4

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Using angles of elevation and depression You see a rock climber on a cliff at a 32° angle of elevation. The horizontal ground distance to the cliff is 1000ft. Find the line of sight distance to the climber.

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A surveyor stands 200 ft from a building to measure its height. The angle of elevation to the top of the building is 35°. How tall is the building?

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An airplane pilot sights a life raft at a 26°angle of depression. The airplane’s altitude is 3km. What is the airplane’s surface distance from the raft?

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Miss Long sits in a treehouse 20ft above the ground. She spots a calculator on the ground with a 15°angle of depression. What is the sight distance between Miss Long and the calculator?

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The world’s tallest unsupported flagpole towers 282 feet above the ground in Surrey, British Columbia. The shortest shadow cast by the flagpole is 137ft long. What is the angle of elevation of the sun when the shadow is cast?

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