# Applied Geometry Geometry Lesson 8 4 Trigonometry Objective: Find trigonometric ratios using right triangles. Use trigonometric ratios to find angle measures in right triangles. Trigonometry Comes from two Greek terms: trigon meaning triangle metron meaning measure Trigonometric Ratio A ratio of the lengths of two sides of a right triangle.

One trig ratio of ABC is AC / AB The triangles are similar by AA similarity. Trig Ratios Sine (written sin) Opposite leg divided by hypotenuse Trig Ratios Cosine (written cos) Adjacent leg divided by hypotenuse Trig Ratios

Tangent (written tan) Opposite leg divided by adjacent leg Express each ratio as a fraction and as a decimal to the nearest hundredth. 15 sin P 0.88 17 8 cos P 0.47 17 15

tan P 1.88 8 8 sin Q 0.47 17 15 cos Q 0.88 17 8 tan Q 0.53 15 Express each ratio as a fraction and as a decimal to the nearest hundredth.

sin J 5 0.38 13 12 cos J 0.92 13 tan J 5 0.42 12 sin K

12 0.92 13 cos K 5 0.38 13 tan K 12 2.4 5

Use special right triangles to express the tangent of 30o as a fraction and as a decimal to the nearest hundredth. tan 30 x x 3 1 3 3 tan 30 3

3 3 Check on calculator: Use special right triangles to express the cosine of 45o as a fraction and as a decimal to the nearest hundredth. cos 45 x x 2 1 2

2 cos 45 2 Check on calculator: Real world A certain part of a hiking trail slopes upward at about a 5 o angle. After traveling a horizontal distance of 100 feet along this part of the trail, what would be the change in a hikers vertical position? What distance has the hiker traveled along the path? x (100) (100) tan 5 100

Use you calculator to take 100 * (tan 5) Be careful order matters how you plug it in on different calculators x 8.75 feet 100 (y) (y) cos 5 y y (cos 5 ) 100 y 100.38 ft cos 5

cos 5 Find x to the nearest hundredth. x tan 25 18 18(tan 25) x x 8.39 15 cos 70 x 15

x cos 70 x 43.86 Inverse Trig Ratios: Used to find Angles Inverse sine Inverse cosine Inverse tangent Inverse ratio

What is tan 30? Approximately 0.58 What is tan-1 0.58? Use the inverse button (usually above the trig button, above tan) Approximately 30o Use a calculator to find the measure of angle A to the nearest tenth. 18 2 sin A or

27 3 2 sin A 3 1 mA 41.8 Use a calculator to find the measure of angle A to the nearest tenth. 6 tan A 20 1 3

tan A 10 mA 16.7 3 cos A 15 1 cos A 5 1 mA 78.5

Solve the right triangle (find all sides and angles). Round side measures to the nearest tenth and angle measures to the nearest degree. 2 2 5 9 ( XY ) 106 XY 10.3 2 5 Use tan instead of sin or tan Y cos since 10.3 is rounded. 9

1 5 tan Y mY 29 9 9 tan X 5 9 tan X 5 1 mX 61

Solve the right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree. mF 23 mG 67 FH = 12.0 mC 28 AB 4.7 BC 8.8 Homework Pg. 567 1 15 all, 16 44 EOE, 62, 66 86 EOE