Circular Trigonometric Functions

Special Angles

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

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

*Special Angles θ 30º 45º 60º sin θ cos θ tan θ 0.7071 0.8660 0.5774 0.5000 0.7071 0.8660 cos θ tan θ 0.5774 1.0000 1.7320

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

sin 300° = - sin 60° = - 0.8660 = cos 300° = cos 60° = 0.5000 = 1/ 2 tan 300° = - tan 60° = -1.732 = csc 300° = - csc 60° = -1.155 = sec 300° = sec 60° = 2.000 = 2/1 cot 300° = - cot 60° = - 0.5774 =

Quadrant Angles

*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)

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

*Quadrant Angles θ 0º 90º 180º 270º sin θ 1 -1 cos θ tan θ

Find trig functions for - 90°. Reference angle is (360° - 90°) → 270° sin 270° = -1 cos 270° = 0 tan 270° undefined csc 270° = -1 sec 270° undefined cot 270° = 0 -90° Use quadrant angle chart. 270°

Coterminal Angles

*Coterminal Angle The angle between 0º and 360º having the same terminal side as a given angle. Ex. 405º - 360º = coterminal angle 45º θ1 = 405º θ2 = 45º

*Coterminal Angles Used with angles greater than 360°, or angles less than 0°. Example cos 900° = cos (900° - 720°) = cos 180° = -1 (See quadrant angles chart)

Example tan (-135° ) = tan (360° -135°) = tan 225° = LOOK→ tan (225° - 180°) tan 45° = 1 (See special angles chart)

Convert from radian to degrees: sec [(7π/ 4)(180/ π)] = sec 315° Find the value of sec 7π / 4 Convert from radian to degrees: sec [(7π/ 4)(180/ π)] = sec 315° SOLUTION

Angle in IV quadrant: sec →positive sec (360° - 315°) = sec 45° = 1 /(cos 45°) = √2 = 1.414 Look how this problem was worked in previous lesson. SOLUTION

Practice

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. sin (- 330°) csc 225° SOLUTION

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

Inverse Trigonometric Functions

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°

More Practice

to the nearest 0.1° for 0° < θ < 360° Given sin θ = 0.3843, find θ to the nearest 0.1° for 0° < θ < 360° SOLUTION

to the nearest 0.1° for 0° < θ < 360° Given cos θ = - 0.0157, find θ to the nearest 0.1° for 0° < θ < 360° SOLUTION

Given sec θ = 1.553 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 0.1° for 0° < θ < 360° SOLUTION

Application

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