Obj: Use tangent to find the length of a side of a right triangle 10.1: Tangent Ratios Obj: Use tangent to find the length of a side of a right triangle

Trigonometric Ratios A ratio of the lengths of 2 sides of a right triangle.

Given an acute angle in a right triangle the 3 sides can be labeled as follows: Leg Opposite Leg Adjacent Hypotenuse

Example a) the leg opposite b) the leg adjacent c) the hypotenuse 1) If using <B state: a) the leg opposite b) the leg adjacent c) the hypotenuse 2) If using <C state:

Supplies: Lined Paper Ruler Protractor

Tangent Ratio For any acute angle A of a right triangle: Tan A= leg opposite <A = a leg adjacent to <A b

Example Find tan S and tan R as fractions in simplified form and as a decimal rounded to 4 decimal places. S 4 4 8 T 4√3 R

Using a calculator for Tangent

Example Approximate the following to 4 decimal places 1) 74 2) 35

Using tangent ratio to Find Leg Length Use a tangent ratio to find the value of x. Round answer to the nearest tenth. 1) 2)

Indirect Measurement FORESTRY You are measuring the height of a Sitka spruce tree in Alaska. You stand 45 feet from the base of a tree. You measure the angle of elevation from a point on the ground to the top of the tree to be 59°. To estimate the height of the tree, you can write a trigonometric ratio that involves the height h and the known length of 45 feet.

A person travels 152 feet on the escalator stairs. Estimating a Distance ESCALATORS The escalator at the Wilshire/Vermont Metro Rail Station in Los Angeles rises 76 feet at a 30° angle. To find the distance d a person travels on the escalator stairs, you can write a trigonometric ratio that involves the hypotenuse and the known leg length of 76 feet. sin 30° = opposite hypotenuse Write ratio for sine of 30°. sin 30° = opposite hypotenuse 76 d Substitute. 30° 76 ft d d sin 30° = 76 Multiply each side by d. d = 76 sin 30° Divide each side by sin 30°. d = 76 0.5 Substitute 0.5 for sin 30°. d = 152 Simplify. A person travels 152 feet on the escalator stairs.