1. 9.5: Trigonometric Ratios

2. Vocabulary Trigonometric Ratio: the ratio of the lengths of two sides of a right triangle Angle of elevation: the angle that your line of sight makes with a line drawn horizontally.

3. Trigonometric Ratios sine A = side opposite hypotenuse cosine A = side adjacent hypotenuse tangent A = side opposite side adjacent AA adjacent o p p o s it e hypotenuse

4. Example 1: Compare the sine, the cosine and the tangent ratios for  A in each triangle below. A A 13 6.5 2.5 5 12 6 sin A = 5/13 sin A =.3846 cos A = 12/13 cos A =.9231 tan A = 5/12 tan A =.4167 sin A = 2.5/6.5 sin A =.3846 cos A = 6/6.5 cos A =.9231 tan A = 2.5/6 tan A =.4167

5. Example 2: Find the sine, cosine, and the tangent of the indicated angle E D F 14 50 48 a)  E sin E = 48/50 cos E = 14/50 tan E = 48/14 = 0.96 =0.28 = 3.4286 b)  D sin D = 14/50 cos D = 48/50 tan D = 14/48 = 0.28 =0.96 = 0.2917

6. Example 3: Find the sine, the cosine, and the tangent of  A 18 18√2 sin A = 18/18√2 cos A = 18/18√2 tan A = 18/18 = 0.7071 = 0.7071 = 1 A

7. Example 4: Find the sine, the cosine, and the tangent of  A 5 10 5√3 sin A = 5/10 cos A = 5√3/10 tan A = 5/5√3 = 0.5 = 0.8660 = 0.5774 A

8. Example 5: Use a calculator to approximate the sine, the cosine, and the tangent of 82 . sin 82  = 0.9903 cos 82  = 0.1392 tan 82  = 7.1154

9. Example 6: You are measuring the height of a building. You stand 100 ft from the base of the building. You measure the angle of elevation from a point on the ground to the top of the building to be 48 . Estimate the height of the building. 100 ft 48  100(tan 48) = h 100(1.1106) = h 111 ft = h The building is about 111 ft tall h

10. Example 7: A driveway rises 12 feet over a distance d at an angle of 3.5 . Estimate the length of the driveway. 12 d d(sin 3.5) =12 0.0610d= 12 d = 196.7213 d = 197 ft