Trigonometry Review Objectives Convert between radians and degrees

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: 0 π/6 π/4 π/3 π/2 0° 30° 45° 60° 90° sinθ 0 1 cosθ 1 0

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: Q 2 Sine ratios are positive in the second quadrant. ALL: Q 1 Sine, Cosine, and Tangent ratios are positive. TAKE: Q 3 Tangent ratios are positive in the third quadrant. CALCULUS: Q 4 Cosine ratios are positive in the fourth quadrant.

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

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) Interval (0, 30) c 20 f(c) + (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=3 sin(4 x) • amplitude • 3 • period • 2 π /4 = π/2 • y=5 tan(6 x) • amplitude • none • period • π/6

Classwork: Trigonometry Review Handout