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Circular Trigonometric Functions

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**Circular Trigonometric Functions**

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

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**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

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**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

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**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

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**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.

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**For any angle in any quadrant**

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

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**NOTE: right-triangle definitions are special case of circular**

functions when θ is in quadrant I Y (x,y) r y θ X x

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***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

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***Both sets of identities are useful to determine trigonometric**

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

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**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)

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**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 (-, -) (+, -)

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Examples

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**#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

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

SOLUTION

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

cot θ = SOLUTION

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

cot θ = θ3 (-4,-3) III SOLUTION

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

cot θ = θ4 (4,-3) IV SOLUTION

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**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

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**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

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**#2 Given: tan θ = -1 and cos θ is positive: **

Draw θ. Show the values for x, y, and r. SOLUTION

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**Given: tan θ = -1 and cos θ is positive: **

Find the six trigonometric functions of θ. SOLUTION

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Calculator Exercise

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**(First determine the reference angle.)**

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

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**(First determine the reference angle.)**

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

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**(First determine the reference angle.)**

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

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Practice

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

SOLUTION

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

SOLUTION

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

SOLUTION

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

SOLUTION

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**#3 Find remaining trigonometric functions:**

sin θ = , tan θ = 1.000 SOLUTION

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**Find remaining trigonometric functions: **

sin θ = , tan θ = 1.000 SOLUTION

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Calculator Practice

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**#1 Express as a function of a reference**

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

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**#2 Express as a function of a reference**

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

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**#3 Find each value on your calculator. (Key in exact angle measure.)**

sin 260.5º tan 150º 10’ SOLUTION

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cot (-240º) csc 450º SOLUTION

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cos 5.41 sec (7/4) SOLUTION

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

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Application

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**# 1 The refraction of a certain prism is**

Calculate the value of n. SOLUTION

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**#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

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Fy = 8.5 lb θ Fx=-4.5 lb SOLUTION

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