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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 . Y-axis. 90º. Quadrant II. Quadrant I. Terminal side. r. r. θ. X-axis.

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## Circular Trigonometric Functions

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**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**Y-axis**90º Quadrant II Quadrant I Terminalside r r θ X-axis 0º 180º Initial side 360º Quadrant III Quadrant IV 270º**Y-axis**-270º Quadrant II Quadrant I -360º X-axis -180º Terminal side Initial side 0º r θ Quadrant III Quadrant IV -90º**angle θ…measured from positive x-axis,**• orinitial 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***Ratio Identities***Both sets of identities are useful to determine trigonometric functions of any angle.**Positive trig values in each quadrant:**Y All all six positive Students sin positive (csc) (-, +) (+, +) II I X III IV Take Classes (-, -) (+, -) tan positive (cot) cos positive (sec)**REMEMBER:**In the ordered pair (x, y), x represents cosine and y represents sine. Y (-, +) (+, +) II I X III IV (-, -) (+, -)**#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 =**y = r = II (-4,3) θ2 sin θ = cos θ = tan θ = csc θ = sec θ = cot θ =**x =**y = r = sin θ = cos θ = tan θ = csc θ = sec θ = cot θ = θ3 (-4,-3) III**x =**y = r = sin θ = cos θ = tan θ = csc θ = sec θ = cot θ = θ4 (4,-3) IV**Y**Perpendicular line from point on circle alwaysdrawn to the x-axis forming a reference triangle II I ref θ2 θ1 X ref θ3 refθ4 IV III**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 refθ4 IV III**#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.)**#1 Draw the angle whose terminal side passes**through the given point.**Find all trigonometric functions for angle whose terminal**side passes thru .**Find all six trigonometric functions:**sin θ = 0.6, cos θ is negative.**#3Find remaining trigonometric functions:**sin θ = - 0.7071, tan θ= 1.000**Find remaining trigonometric functions:**sin θ = - 0.7071, tan θ= 1.000**Calculator**Practice**#1 Express as a function of a reference**angle and find the value: cot 306º .**#2 Express as a function of a reference angle and find**the value: sec (-153º) .**#3 Find each value on your calculator.**(Key in exact angle measure.) sin 260.5º tan 150º 10’**cot (-240º)**csc 450º**cos 5.41**sec (7/4)**π/2 = 1.57**0 2π = 6.28 π = 3.14 3π/2 = 4.71**# 1 The refraction of a certain prism is**Calculate the value of n.**#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**Fy = 8.5 lb**θ Fx=-4.5 lb

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