Trigonometric Functions of Any Angles Objective: To evaluate trig functions of any angle by using reference angles

(x, y) θ r y r sin θ = csc θ = r y x r cos θ = sec θ = r x y x tan θ = cot θ = x y

Evaluate the six trigonometric functions if the point (-4,3) lies on the terminal side of an angle θ. 3 5 sin θ = csc θ = 5 3 -4 5 cos θ = sec θ = 5 -4 3 -4 tan θ = cot θ = -4 3

Evaluate the six trigonometric functions if the point (1,-3) lies on the terminal side of an angle θ. sin θ = csc θ = cos θ = sec θ = tan θ = cot θ =

Evaluate the six trigonometric functions if the point (-7,-4) lies on the terminal side of an angle θ. sin θ = csc θ = cos θ = sec θ = tan θ = cot θ =

Evaluate the six trigonometric functions if the point (3, 0) lies on the terminal side of an angle θ. sin θ = csc θ = cos θ = sec θ = tan θ = cot θ =

When Trig Functions are +/- sin, csc + cos, sec – tan, cot – sin, csc + cos, sec + tan, cot + II I (-,+) (+,+) III IV sin, csc – cos, sec – tan, cot + sin, csc – cos, sec + tan, cot – (-,-) (+,-)

Name the quadrant in which the angle lies. sinθ < 0 and cosθ < 0 sinθ > 0 and tanθ < 0 III II

Evaluate the six trigonometric functions if the point (0, 3) lies on the terminal side of an angle θ. sin θ = csc θ = cos θ = sec θ = tan θ = cot θ =

Evaluate the six trigonometric functions if the point (-3, 0) lies on the terminal side of an angle θ. sin θ = csc θ = cos θ = sec θ = tan θ = cot θ =

Evaluate the six trigonometric functions if the point (0, -3) lies on the terminal side of an angle θ. sin θ = csc θ = cos θ = sec θ = tan θ = cot θ =

If (3,0) is on the terminal side of the angle, then θ = 0°. 90° (0,3) (3,0) If (3,0) is on the terminal side of the angle, then θ = 0°. If (0,3) is on the terminal side of the angle, then θ = 90°

If (-3,0) is on the terminal side of the angle, then θ = 180° 270° (-3,0) (0,-3) If (-3,0) is on the terminal side of the angle, then θ = 180° If (0,-3) is on the terminal side of the angle, then θ = 270°

Quadrantal Angle Angle whose terminal side falls on an axis. Examples: 0°, 90°, 180°, 270°, 360°

Trig Functions of Quadrantal Angles θ sin cos tan csc sec cot 0° 90° 180° 270° 360°

Reference Angles Remember: Must be positive Must be acute The acute angle formed by the terminal side of the angle and the horizontal axis (x-axis). Remember: Must be positive Must be acute

Find the reference angle. 140° 180° – 140° = 40°

Find the reference angle. 230° 230° – 180° = 50°

Short-Cuts for Reference Angles Quadrant θ in degrees θ in radians I II III IV If θ goes around more than once… θ θ 180° - θ π - θ θ - 180° θ - π 360° - θ 2π - θ θ - 360°. Then proceed with above. θ - 2π Then proceed with above.

Find the reference angle. 310 170 305 360° - 310° = 50° 180° - 170 ° = 10° 360° - 305° = 55°

Find the reference angle.

Evaluate the following. 225° - 180° = 45° cos(45°) Quadrant III In Quadrant III, cos is negative. 180° - 120° = 60° Quadrant II tan(60°) In Quadrant II, tan is negative.

Evaluate the following. Quadrant III In Quadrant III, tan is positive.

Evaluate the following. Quadrant III In Quadrant III, sin is negative.

Find the reference angle. 375 470 595 375° - 360° = 15° 470° - 360 ° = 110° 180° - 110 ° = 70° 595° - 360° = 235° 235° - 180 ° = 55°

Evaluate the following. It’s larger than 360°! 390° - 360° = 30° Quadrant I. sin(30°) In Quadrant I, sin is positive.

Evaluate the following. It’s larger than 2π! Quadrant I!

If θ is negative… Add 360° (or 2π) until you get a positive angle. Proceed as usual.

Find the reference angle. -275 -190 -112 -275° + 360° = 85° -190° + 360 ° = 170° 180° - 170° = 10° -112° + 360° = 248° 248° - 180 ° = 68°

Find the reference angle. Quadrant III.

Evaluate the following. It’s negative. -120 + 360 = 240° Quadrant III. 240° - 180° = 60° sin(60°) In Quadrant III, sin is negative.