Trigonometry Review

Objectives: Convert between radians and degrees. Calculate values of trigonometric functions. Apply trig identities in solving trigonometric equations and inequalities.

Radians: When stating solutions to trigonometric equations in Calculus we use: RADIANS To convert from degrees to radians: multiply by π / 180 To convert from radians to degrees: multiply by 180 / π

Examples: Change 120° to radians 120° × π /180° =2π/3 Change π/10 to degrees π /10 × 180° / π = 18°

Reciprocal and Quotient Identities: You will be expected to KNOW these !

Basic Trig Identities: You will be expected to KNOW these !

Deriving Other Identities:

Deriving Other Identities:

Deriving Other Identities:

Deriving Other Identities:

Quadrant I Unit Circle Values: π/6 π/4 π/3 π/2 0° 30° 45° 60° 90° sinθ 1 cosθ

Using Reference Angles: The second quadrant reference angle occurs at 180°- Ɵ° If the first quadrant angle is Ɵ° The third quadrant reference angle occurs at 180°+ Ɵ° The fourth quadrant reference angle occurs at 360°- Ɵ° The first quadrant reference angle is the positive acute angle from a given angle to the x-axis. Each Trig ratio’s magnitude is the same for all four reference angles, while the signs of the corresponding trig ratios may differ.

ALL STUDENTS TAKE CALCULUS An easy trick for remembering which quadrant a given trig ratio is positive in can be seen as follows: STUDENTS: Q2 Sine ratios are positive in the second quadrant. ALL: Q1 Sine, Cosine, and Tangent ratios are positive. TAKE: Q3 Tangent ratios are positive in the third quadrant. CALCULUS: Q4 Cosine ratios are positive in the fourth quadrant.

Evaluating a Trig Ratio: Example: Using reference angles is quicker than the Unit Circle! Thought Process:

The Unit Circle:

Solving Trig Equations/Inequalities: Before solving you MUST check for domain restrictions. Problems set equal to zero often solve by factoring. Substitute in appropriate trig identities to help solve the given equation. It is helpful to manipulate equations to be entirely in terms of sin or cosine when possible. When solving trig inequalities test intervals on [0,2π) considering solutions and restrictions.

Example 1: solve over [0,2π)

Example 2: solve the equation over [0,2π).

Example 3: solve on [0,2π) [30°,180°) U [210°,360°) Interval c f(c) (0,30) 20 + (30,180) 90 - (180,210) 200 (210,360) 300 [30°,180°) U [210°,360°)

Period and Amplitude: y=a f (bx) where f is a trig function amplitude: |a| if it has an amplitude (sine and cosine) period: 2 π/ |b| for sine, cosine, secant and cosecant π/ |b| for tangent and cotangent

Examples: y=3sin(4x) amplitude 3 period 2 π /4 = π/2 y=5tan(6x) none π/6

Trigonometry Review Handout Classwork: Trigonometry Review Handout