*Special Angles 45° 60° 30° 30°, 45°, and 60° → common reference angles Memorize their trigonometric functions. Use the Pythagorean Theorem; triangles below.

1 2 *Special Angles 60° 30° 30°, 45°, and 60° → common reference angles Memorize their trigonometric functions. Use the Pythagorean Theorem; triangles below.

1 1 *Special Angles 45° 30°, 45°, and 60° → common reference angles Memorize their trigonometric functions. Use the Pythagorean Theorem; triangles below.

* Special Angles θ 0º 30º 45º 60º 90º sin θ cos θ tan θ

RECIPROCAL TRIG FUNCTIONS SIN  CSC  COS  SEC  TAN  COT 

RECIPROCAL TRIG FUNCTIONS SIN  CSC  COS  SEC  TAN  COT 

Find trig functions of 300° without calculator. Reference angle is 60°[360°  300°]; IV quadrant 300° 60° sin 300° = cos 300° = tan 300° = csc 300° = sec 300° = cot 300° = Use special angle chart.

Find trig functions of 120° without calculator. Reference angle is 120°[180°  120°]; IIquadrant sin 120° = cos 120° = tan 120° = csc 120° = sec 120° = cot 120° = Use special angle chart. 60°

Find trig functions of 210° without calculator. Reference angle is [210°-180°]; III quadrant 60° sin 210° = cos 210° = tan 210° = csc 210° = sec 210° = cot 210° = Use special angle chart.

sin 300° = cos 300° = tan 300° = csc 300° = sec 300° = cot 300° =

12*Quadrant Angles Reference angles cannot be drawn for quadrant angles 0°, 90°, 180°, and 270° Determined from the unit circle; r = 1 Coordinates of points (x, y) correspond to (cos θ, sin θ)

0° (1,0) 180° (  1,0) *Quadrant Angles 90° (0,1) → (cos θ, sin θ) 270° (0,  1) r = 1

*Quadrant Angles θ 0º 90º 180º 270º sin θ cos θ tan θ 0 0

Find trig functions for  90°. Reference angle is (360°  90°) → 270° sin 270° = cos 270° = tan 270°= csc 270° = sec 270°= cot 270° = 270°  90° Use quadrant angle chart.

*Coterminal Angle θ 1 = 405º The angle between 0º and 360º having the same terminal side as a given angle. Ex. 405º  360º = coterminal angle 45º θ 2 = 45º

*Coterminal Angles Example cos 900° = (See quadrant angles chart) Used with angles greater than 360°, or angles less than 0°.

Example tan (  135° ) = (See special angles chart)

Find the value of sec 7π / 4 SOLUTION

Express as a function of the reference angle and find the value. tan 210°sec 120 ° SOLUTION

Express as a function of the reference angle and find the value. csc 225° sin (  330°) SOLUTION

Express as a function of the reference angle and find the value. cos (  5π) cot (9π/2) SOLUTION

SINCOSTANSECCSCCOT

Inverse Trig Functions Used to find the angle when two sides of right triangle are known... or if its trigonometric functions are known Notation used: Read: “angle whose sine is …” Also,

Inverse trig functions have only one principal value of the angle defined for each value of x:  90° < arcsin < 90° 0° < arccos < 180°  90° < arctan < 90°

Example: Given tan θ =  1.600, find θ to the nearest 0.1° for 0° < θ < 360° Tan is negative in II & IV quadrants

θ = 180°  58.0° = 122° II θ = 360°  58.0° = 302° IV Note: On the calculator entering results in  58.0°

Given sin θ = , find θ to the nearest 0.1° for 0° < θ < 360° SOLUTION

Given cos θ =  , find θ to the nearest 0.1° for 0° < θ < 360° SOLUTION

Given sec θ = where sin θ < 0, find θ to the nearest 0.1° for 0° < θ < 360° SOLUTION

Given the terminal side of θ passes through point (2, -1), find θ the nearest tenth for 0° < θ < 360° SOLUTION

Given the terminal side of θ passes through point (3, 5), find θ the nearest tenth for 0° < θ < 360° SOLUTION

Given the terminal side of θ passes through point (8, 8), find θ the nearest tenth for 0° < θ < 360° SOLUTION

Given the terminal side of θ passes through point (-5, 12), find θ the nearest tenth for 0° < θ < 360° SOLUTION

The voltage of ordinary house current is expressed as V = 170 sin 2πft, where f = frequency = 60 Hz and t = time in seconds. Find the angle 2πft in radians when V = 120 volts and 0 < 2πft < 2π SOLUTION

Find t when V = 120 volts SOLUTION

The angle β of a laser beam is expressed as: where w = width of the beam (the diameter) and d = distance from the source. Find β if w = 1.00m and d = 1000m. SOLUTION