Circular Trigonometric Functions

Circular Trigonometric Functions Y circle…center at (0,0) radius r…vector with length/direction r θ X angle θ… determines direction

Quadrant II Quadrant I 360º Quadrant III Quadrant IV Y-axis 90º r r θ Terminal side r r θ 0º X-axis 180º Initial side 360º Quadrant III Quadrant IV 270º

Quadrant II Quadrant I Quadrant III Quadrant IV Y-axis -270º -360º X-axis -180º Terminal side Initial side 0º r θ Quadrant III Quadrant IV -90º

angle θ…measured from positive x-axis, or initial side, to terminal side counterclockwise: positive direction clockwise: negative direction four quadrants…numbered I, II, III, IV counterclockwise

six trigonometric functions for angle θ whose terminal side passes thru point (x, y) on circle of radius r sin θ = y / r csc θ = r / y cos θ = x / r sec θ = r / x tan θ = y / x cot θ = x / y These apply to any angle in any quadrant.

For any angle in any quadrant x2 + y2 = r2 … So, r is positive by Pythagorean theorem. (x,y) r y θ x

NOTE: right-triangle definitions are special case of circular functions when θ is in quadrant I Y (x,y) r y θ X x

*Reciprocal Identities sin θ = y / r and csc θ = r / y cos θ = x / r and sec θ = r / x tan θ = y / x and cot θ = x / y

*Both sets of identities are useful to determine trigonometric *Ratio Identities *Both sets of identities are useful to determine trigonometric functions of any angle.

Students Take Classes Positive trig values in each quadrant: All Y Students All all six positive sin positive (csc) (-, +) (+, +) II I X III IV Take Classes (-, -) (+, -) tan positive (cot) cos positive (sec)

In the ordered pair (x, y), x represents cosine and REMEMBER: In the ordered pair (x, y), x represents cosine and y represents sine. Y (-, +) (+, +) II I X III IV (-, -) (+, -)

Examples

#1 Draw each angle whose terminal side passes through the given point, and find all trigonometric functions of each angle. θ1: (4, 3) θ2: (- 4, 3) θ3: (- 4, -3) θ4: (4, -3)

x = y = r = I (4,3) sin θ = cos θ = tan θ = csc θ = sec θ = cot θ = θ1

x = II y = r = (-4,3) θ2 sin θ = cos θ = tan θ = csc θ = sec θ = cot θ =

x = y = r = θ3 (-4,-3) III sin θ = cos θ = tan θ = csc θ = sec θ = cot θ = θ3 (-4,-3) III

x = y = r = θ4 (4,-3) IV sin θ = cos θ = tan θ = csc θ = sec θ = cot θ = θ4 (4,-3) IV

#2 Given: tan θ = -1 and cos θ is positive: Draw θ. Show the values for x, y, and r.

Given: tan θ = -1 and cos θ is positive: Find the six trigonometric functions of θ.

Calculator Exercise

# 1 Find the value of sin 110º. (First determine the reference angle.)

#2 Find the value of tan 315º. (First determine the reference angle.)

#3 Find the value of cos 230º. (First determine the reference angle.)

Practice

#1 Draw the angle whose terminal side passes through the given point .

Find all trigonometric functions for angle whose terminal side passes thru .

#2 Draw angle: sin θ = 0.6, cos θ is negative.

Find all six trigonometric functions: sin θ = 0.6, cos θ is negative.

#3 Find remaining trigonometric functions: sin θ = - 0.7071, tan θ = 1.000

Find remaining trigonometric functions: sin θ = - 0.7071, tan θ = 1.000