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

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Quadrant IQuadrant II Quadrant IIIQuadrant IV Y-axis X-axis 90º 0º 180º 270º Initial side Terminal side θ rr 360º

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Quadrant IQuadrant II Quadrant IIIQuadrant IV Y-axis X-axis -270º -360 º -180º -90º Initial side θ r 0º0º Terminal side

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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 / rsec θ = r / x tan θ = y / xcot θ = x / y These apply to any angle in any quadrant.

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For any angle in any quadrant x 2 + y 2 = r 2 … So, r is positive by Pythagorean theorem. r θ x y (x,y)

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

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sin θ = y / r and csc θ = r / y cos θ = x / randsec θ = r / x tan θ = y / xandcot θ = x / y *Reciprocal Identities

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

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Y X (-, +) (+, +) (-, -)(+, -) Positive trig values in each quadrant: A ll all six positive S tudents T ake C lasses sin positive (csc) tan positive (cot) cos positive (sec) I III II IV

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

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#1 Draw each angle whose terminal side passes through the given point, and find all trigonometric functions of each angle. θ 1 : (4, 3) θ 2 : (- 4, 3) θ 3 : (- 4, -3) θ 4 : (4, -3) SOLUTION

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I x = y = r = (4,3) sin θ = cos θ = tan θ = csc θ = sec θ = cot θ = θ1θ1 SOLUTION

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II x = y = r = (-4,3) sin θ = cos θ = tan θ = csc θ = sec θ = cot θ = θ2θ2 SOLUTION

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III x = y = r = (-4,-3) sin θ = cos θ = tan θ = csc θ = sec θ = cot θ = θ3θ3 SOLUTION

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IV x = y = r = (4,-3) sin θ = cos θ = tan θ = csc θ = sec θ = cot θ = θ4θ4 SOLUTION

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Y X θ1θ1 I II III IV ref θ 2 ref θ 3 ref θ 4 Perpendicular line from point on circle always drawn to the x-axis forming a reference triangle

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Y X θ1θ1 I II III IV ref θ 2 ref θ 3 ref θ 4 Value of trig function of angle in any quadrant is equal to trig function of its reference angle, or it differs only in sign.

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

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

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

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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 = - 0.7071, tan θ = 1.000 SOLUTION

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

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#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 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 260.5ºtan 150º 10’ SOLUTION

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csc 450ºcot (-240º) SOLUTION

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sec (7 /4) cos 5.41 SOLUTION

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π/2 = 1.57 π = 3.14 3π/2 = 4.71 0 2π = 6.28

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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 F x = - 4.5 lb and F y = 8.5 lb. Find sin θ and cos θ. F y = 8.5 lb F x =-4.5 lb θ SOLUTION

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F y = 8.5 lb F x =-4.5 lb θ SOLUTION

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