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TRIGONOMETRY

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Presentation on theme: "TRIGONOMETRY"— Presentation transcript:

1 TRIGONOMETRY

2 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.

3 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

4 Special Right Triangles 30° 45° 60°45°

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

6 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

7 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

8 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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10 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

11 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)

12 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)

13 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

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

15 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)

16 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.

17 Trigonometric Identities Summation & Difference Formulas

18 Trigonometric Identities Double Angle Formulas

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

20 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.

21 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

22 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) = – = 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

23 Acknowledgements Unit Circle: Text: Blitzer, Precalculus Essentials, Pearson Publishing, 2006.


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