4.3 Right Triangle Trigonometry

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” hypotenuse Side oppositeθ θ Side adjacent to θ

Trigonometric Functions • Let θ be an acute angle of a right triangle. RECIPROCALS

Evaluating Trig Functions • Use the triangle to find the exact values of the six trig functions of θ. hypotenuse 4 θ 3

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

Evaluating Trig Functions for 45° • Find the exact value of sin 45°, cos 45°, and tan 45°

Evaluating Trig Functions for 30°and 60° • Find the exact values of sin60°, cos 60°, sin 30°, cos 30° 60° 30°

Sine, Cosine, and Tangent of Special Angles

sin30° = ½ = cos60° (notice that 30° and 60° are complementary angles) sin(90° - θ) = cos θ cos(90° - θ) = sin θ tan(90° - θ) = cot θ cot(90° - θ) = tan θ sec(90° - θ) = csc θ csc(90° - θ) = sec θ

Trig Identities • Reciprocal Identities

Trig Identities (cont) • Quotient Identities • Pythagorean Identities

Applying Trig Identities • Let θ be an acute angle such that sin θ = .6. Find the values of (a) cosθ and (b) tan θ using trig identities.

Using Trig Identities • Use trig identities to transform one side of the equation into the other (0 < θ < π/2) • cosθ sec θ = 1 b) (sec θ + tan θ)(secθ – tanθ) = 1

Evaluating Using the Calculator • sin 63° • tan (36°) • sec (5°)

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 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.

4.3 Right Triangle Trigonometry