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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 Side opposite θ hypotenuse θ Using the lengths of these 3 sides, we form six ratios that define the six trigonometric functions of the acute angle θ. sine cosine tangent cosecant secant cotangent Side adjacent to θ *notice each pair has a “co” Trigonometric Functions • Let θ be an acute angle of a right triangle. sin cos tan hypotenuse RECIPROCALS hyp csc opp hyp sec adj adj cot opp opposite θ adjacent Evaluating Trig Functions • Use the triangle to find the exact values of the six trig functions of θ. sin 𝜃 = csc 𝜃 = cos 𝜃 = sec 𝜃 = tan 𝜃 = cot 𝜃 = 5 θ 4 3 • sin 𝛽 = csc 𝛽 = cos 𝛽 = sec 𝛽 = tan 𝛽 = cot 𝛽 = Repeat for β. What do you notice? Solving Right Triangles with Trig • Use the trig functions to find the missing sides for θ = 57° 5 h θ θ 2. Use the trig functions to find the missing sides for θ = 32° 1. h 1. • a 17 θ 2. 2 x θ 3. θ 27 3. x o θ x 3 100 Special Right Triangles 45-45-90 30-60-90 45° 60° 2 1 2 1 45° 1 30° 3 Evaluating Trig Functions for 45° • Find the exact value of the following: • sin 45° = • cos 45° = 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° = 30° 60° Sine, Cosine, and Tangent of Special Angles Θ in Degrees 30° π 6 60° π 3 45° π 4 sin Θ 3 2 1 2 2 2 cos Θ 1 2 3 2 2 2 tan Θ 3 3 3 1 Θ in Radians 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. 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.