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) SOLUTION

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

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

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

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

Perpendicular II I line from point on circle always drawn to the x-axis forming a reference triangle II I ref θ2 θ1 X ref θ3 ref θ4 III IV

is equal to trig function of its reference angle, or it differs Value of trig function of angle in any quadrant is equal to trig function of its reference angle, or it differs only in sign. Y II I ref θ2 θ1 X ref θ3 ref θ4 III IV

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

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

Calculator Exercise

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

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

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

Practice

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

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

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

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

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

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

Calculator Practice

#1 Express as a function of a reference #1 Express as a function of a reference angle and find the value: cot 306º . SOLUTION

#2 Express as a function of a reference #2 Express as a function of a reference angle and find the value: sec (-153º) . SOLUTION

#3 Find each value on your calculator. (Key in exact angle measure.) sin 260.5º tan 150º 10’ SOLUTION

cot (-240º) csc 450º SOLUTION

cos 5.41 sec (7/4) SOLUTION

π/2 = 1.57 2π = 6.28 π = 3.14 3π/2 = 4.71

Application

# 1 The refraction of a certain prism is Calculate the value of n. SOLUTION

#2 A force vector F has components Fx = - 4.5 lb and Fy = 8.5 lb. Find sin θ and cos θ. Fy = 8.5 lb θ Fx=-4.5 lb SOLUTION

Fy = 8.5 lb θ Fx=-4.5 lb SOLUTION