4.3 Right Triangle Trigonometry Objectives: Evaluate trigonometric functions of acute angles Use trig identities Evaluate trig functions with a calculator Use trig functions to model and solve real life problems

Right Triangle Trigonometry Using the lengths of these 3 sides, we form six ratios that define the six trigonometric functions of the acute angle θ. sine cosecant cosine secant tangent cotangent *notice each pair has a “co” Side opposite θ hypotenuse θ Side adjacent to θ

Trigonometric Functions Let θ be an acute angle of a right triangle. hypotenuse RECIPROCALS opposite θ adjacent

Evaluating Trig Functions Use the triangle to find the exact values of the six trig functions of θ. Repeat for β. What do you notice? sin 𝜃= csc 𝜃= cos 𝜃= sec 𝜃= tan 𝜃= cot 𝜃= sin 𝛽= csc 𝛽= cos 𝛽= sec 𝛽= tan 𝛽= cot 𝛽= 5 4 θ 3

Solving Right Triangles with Trig Use the trig functions to find the missing sides for θ = 57° Use the trig functions to find the missing sides for θ = 32° h 5 θ h 17 x θ 2 x θ θ a θ 27 o x 100 3 θ

Special Right Triangles 45-45-90 30-60-90 45° 60° 2 1 2 1 45° 30° 1 3

Evaluating Trig Functions for 45° Find the exact value of the following: sin 45° = cos 45° = tan 45° = 45°

Evaluating Trig Functions for 30° and 60° Find the exact values of the following: sin 60° = • sin 30° = cos 60° = • cos 30° = tan 60° = • tan 30° = 60°

Sine, Cosine, and Tangent of Special Angles Θ in Degrees 30° 60° 45° Θ in Radians π 6 π 3 π 4 sin Θ 3 2 1 2 2 2 cos Θ tan Θ 3 3 3 1

Applications of Right Triangle Trigonometry Angle of elevation: the angle from the horizontal upward to the object Angle of depression: the angle from the horizontal downward to the object

Word Problems A surveyor is standing 50 feet from the base of a large tree. The surveyor measure the angle of elevation to the top of the tree as 71.5°. How tall is the tree?

You are 200 yards from a river You are 200 yards from a river. Rather than walk directly to the river, you walk 400 yards along a straight, diagonal path to the river’s edge. Find the acute angle θ between this path and the river’s edge.

Find the length c of the skateboard ramp.