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TRIGONOMETRY http://math.la.asu.edu/~tdalesan/mat170/TRIGONOMETRY.ppt

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Angles, Arc length, Conversions Angle measured in standard position. Initial side is the positive x – axis which is fixed. Terminal side is the ray in quadrant II, which is free to rotate about the origin. Counterclockwise rotation is positive, clockwise rotation is negative. Coterminal Angles: Angles that have the same terminal side. 60°, 420°, and –300° are all coterminal. Degrees to radians: Multiply angle by radians Radians to degrees: Multiply angle by Arc length = central angle x radius, or Note: The central angle must be in radian measure. Note: 1 revolution = 360° = 2π radians.

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Right Triangle Trig Definitions sin(A) = sine of A = opposite / hypotenuse = a/c cos(A) = cosine of A = adjacent / hypotenuse = b/c tan(A) = tangent of A = opposite / adjacent = a/b csc(A) = cosecant of A = hypotenuse / opposite = c/a sec(A) = secant of A = hypotenuse / adjacent = c/b cot(A) = cotangent of A = adjacent / opposite = b/a A a b c B C

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Special Right Triangles 30° 45° 60°45° 2 1 1 1

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Basic Trigonometric Identities Quotient identities: Reciprocal Identities: Pythagorean Identities: Even/Odd identities: Even functionsOdd functions

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ASTC All Students Take Calculus. Quad II Quad I Quad III Quad IV cos(A)>0 sin(A)>0 tan(A)>0 sec(A)>0 csc(A)>0 cot(A)>0 cos(A)<0 sin(A)>0 tan(A)<0 sec(A)<0 csc(A)>0 cot(A)<0 cos(A)<0 sin(A)<0 tan(A)>0 sec(A)<0 csc(A)<0 cot(A)>0 cos(A)>0 sin(A)<0 tan(A)<0 sec(A)>0 csc(A)<0 cot(A)<0

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Reference Angles Quad I Quad II Quad III Quad IV θ’ = θθ’ = 180° – θ θ’ = θ – 180°θ’ = 360° – θ θ’ = π – θ θ’ = 2π – θ θ’ = θ – π

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Unit circle Radius of the circle is 1. x = cos(θ) y = sin(θ) Pythagorean Theorem: This gives the identity: Zeros of sin(θ) are where n is an integer. Zeros of cos(θ) are where n is an integer.

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Graphs of sine & cosine Fundamental period of sine and cosine is 2π. Domain of sine and cosine is Range of sine and cosine is [–|A|+D, |A|+D]. The amplitude of a sine and cosine graph is |A|. The vertical shift or average value of sine and cosine graph is D. The period of sine and cosine graph is The phase shift or horizontal shift is

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Sine graphs y = sin(x) y = sin(3x) y = 3sin(x) y = sin(x – 3) y = sin(x) + 3 y = 3sin(3x-9)+3 y = sin(x) y = sin(x/3)

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Graphs of cosine y = cos(x) y = cos(3x) y = cos(x – 3) y = 3cos(x) y = cos(x) + 3 y = 3cos(3x – 9) + 3 y = cos(x) y = cos(x/3)

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Tangent and cotangent graphs Fundamental period of tangent and cotangent is π. Domain of tangent is where n is an integer. Domain of cotangent where n is an integer. Range of tangent and cotangent is The period of tangent or cotangent graph is

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Graphs of tangent and cotangent y = tan(x) Vertical asymptotes at y = cot(x) Verrical asymptotes at

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Graphs of secant and cosecant y = sec(x) Vertical asymptotes at Range: (–∞, –1] U [1, ∞) y = cos(x) y = csc(x) Vertical asymptotes at Range: (–∞, –1] U [1, ∞) y = sin(x)

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Inverse Trigonometric Functions and Trig Equations Domain: [–1, 1] Range: 0 < y < 1, solutions in QI and QII. –1 < y < 0, solutions in QIII and QIV. Domain: [–1, 1] Range: [0, π] 0 < y < 1, solutions in QI and QIV. –1< y < 0, solutions in QII and QIII. Domain: Range: 0 < y < 1, solutions in QI and QIII. –1 < y < 0, solutions in QII and QIV.

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Trigonometric Identities Summation & Difference Formulas

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Trigonometric Identities Double Angle Formulas

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Trigonometric Identities Half Angle Formulas The quadrant of determines the sign.

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Law of Sines & Law of Cosines Law of sinesLaw of cosines Use when you have a complete ratio: SSA. Use when you have SAS, SSS.

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Vectors A vector is an object that has a magnitude and a direction. Given two points P1: and P2: on the plane, a vector v that connects the points from P1 to P2 is v = i + j. Unit vectors are vectors of length 1. i is the unit vector in the x direction. j is the unit vector in the y direction. A unit vector in the direction of v is v/||v|| A vector v can be represented in component form by v = v x i + v y j. The magnitude of v is ||v|| = Using the angle that the vector makes with x-axis in standard position and the vector’s magnitude, component form can be written as v = ||v||cos(θ)i + ||v||sin(θ)j

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Vector Operations Scalar multiplication: A vector can be multiplied by any scalar (or number). Example: Let v = 5i + 4j, k = 7. Then kv = 7(5i + 4j) = 35i + 28j. Dot Product: Multiplication of two vectors. Let v = v x i + v y j, w = w x i + w y j. v · w = v x w x + v y w y Example: Let v = 5i + 4j, w = –2i + 3j. v · w = (5)(–2) + (4)(3) = –10 + 12 = 2. Two vectors v and w are orthogonal (perpendicular) iff v · w = 0. Addition/subtraction of vectors: Add/subtract same components. Example Let v = 5i + 4j, w = –2i + 3j. v + w = (5i + 4j) + (–2i + 3j) = (5 – 2)i + (4 + 3)j = 3i + 7j. 3v – 2w = 3(5i + 4j) – 2(–2i + 3j) = (15i + 12j) + (4i – 6j) = 19i + 6j. ||3v – 2w|| = Alternate Dot Product formula v · w = ||v||||w||cos(θ). The angle θ is the angle between the two vectors. θ w v

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Acknowledgements Unit Circle: http://www.davidhardison.com/math/trig/unit_circle.gif Text: Blitzer, Precalculus Essentials, Pearson Publishing, 2006.

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