60º 5 ? 45º 8 ? Recall: How do we find “?”

65º 5 ? What about this one?

60º 5 ? What is the ratio of long leg to short leg? 60º 11 ? 60º 7 ?

65º 5 ? 12 ? 65º 123 ? These triangles are all similar (AA~). What is the relationship of their ratios of long leg to short leg? The ratios are all the same.

Right Triangle Trigonometry Sections 9.1 and 9.2

What is Trigonometry? Angle Problem Triangle Sum Theorem Side Problem Pythagorean Theorem Angle and Side Problem

Tangent Ratio Adjacent Leg Opposite Leg B C A Adjacent Leg Opposite Leg B C A Trig ratios are always with respect to a specific angle.

Labeling in a right triangle a b B C A c

tan A === opposite adjacent BC AC tan B === opposite adjacent AC BC Write the tangent ratios for A and B.

Calculator Trig Functions 37  B C A Make sure the calculator  is set to “degrees” If you must round, use at least 3 decimal places.

To measure the height of a tree, Alma 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. The tree forms a right angle with the ground, so you can use the tangent ratio to estimate the height of the tree. tan 32° = height 125 Use the tangent ratio. height = 125 (tan 32°)Solve for height Use a calculator. The tree is about 78 ft tall.

Sine Ratio Opposite Leg Hypotenuse B C A Opposite Leg Hypotenuse B C A

Cosine Ratio Adjacent Leg Hypotenuse B C A Adjacent Leg Hypotenuse B C A

Use the triangle to find sin T, cos T, sin G, and cos G. Write your answer in simplest terms. sin T == = opposite hypotenuse cos T == = adjacent hypotenuse sin G == = opposite hypotenuse cos G == = adjacent hypotenuse

Calculator Trig Functions 37  B C A Make sure the calculator  is set to “degrees”

A 20-ft. wire supporting a flagpole forms a 35˚ angle with the flagpole. To the nearest foot, how high is the flagpole? The flagpole, wire, and ground form a right triangle with the wire as the hypotenuse. Because you know an angle and the measures of its adjacent side and the hypotenuse, you can use the cosine ratio to find the height of the flagpole. cos 35° = height 20 Use the cosine ratio. height = 20 cos 35°Solve for height Use a calculator. The flagpole is about 16 ft tall.

SOH-CAH-TOA SOH CAH TOA SOH-CAH-TOA

Inverse Trig Functions x  B C A If the Sin of an angle is , what is the measure of the angle?

Regular vs. Inverse

A right triangle has a leg 1.5 units long and hypotenuse 4.0 units long. Find the measures of its acute angles to the nearest degree. Draw a diagram using the information given. Use the inverse of the cosine function to find m A. cos A = =Use the cosine ratio. Use the inverse of the cosine.m A = cos –1 (0.375) Use a calculator Round to the nearest degree.m A 68

(continued) To find m B, use the fact that the acute angles of a right triangle are complementary. The acute angles, rounded to the nearest degree, measure 68 and 22. m A + m B = 90Definition of complementary angles Substitute.68 + m B 90 m B 22

Find m R to the nearest degree. tan R = Find the tangent ratio. So m R 49. m R tan –1 Use the inverse of the tangent Use a calculator