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Ch. 5 Trigonometric Functions of Real Numbers

Ch. 5 Trigonometric Functions of Real Numbers. Melanie Kulesz Katie Lariviere Austin Witt. The Unit Circle. x 2 + y 2 = 1. The circle of radius 1 centered at the origin in the xy -plane . Proving points on the unit circle. Use equation: x2 + y2 = 1 See Example. See example

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Ch. 5 Trigonometric Functions of Real Numbers

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  1. Ch. 5 Trigonometric Functions of Real Numbers Melanie Kulesz Katie Lariviere Austin Witt

  2. The Unit Circle x2 + y2 = 1 The circle of radius 1 centered at the origin in the xy-plane.

  3. Proving points on the unit circle Use equation: x2 + y2 = 1 See Example

  4. See example Use equation: x2 + y2 = 1 Locating a point on the circle

  5. Terminal Points • Terminal Point – the point P(x,y) obtained and determined by the real number t • Suppose t is a real number. Mark off a distance t along the unit circle, starting at the point (1,0) and moving in a counterclockwise direction if t is positive or in a clockwise direction if t is negative • See example t = -π

  6. Reference Numbers • Reference Number - the shortest distance along the unit circle between the terminal point determined by t and the x-axis • Sine Curve • Cosine Curve • Tangent Curve • Stretch • Shift • Amplitude • Period

  7. Using Reference Numbers to Find Terminal Points To find the terminal point P determined by any value of t, use the following steps… • Find the reference number t • Find the terminal point Q(a, b) determined by t • The terminal point determined by t is P(±a, ±b ), where the signs are chosen according to the quadrant in which this terminal point lies See Example

  8. Trigonometric Functions sin t = y cos t = x tan t = y/x (x≠0) csc t = 1/y (y≠0) sec t = 1/x (x≠0) cot t = x/y (y≠0) See example

  9. Even-Odd Properties • Sin(-t) = -sin t • Cos(-t) = cos t • Tan(-t) = -tan t • Csc(-t) = -csc t • Sec(-t) = sec t • Cot(-t) = -cot t Odd Even

  10. QuadrantPositive FunctionsNegative functions Iallnone IIsin, csccos, sec, tan, cot IIItan, cotsin, csc, cos, sec IVcos, secsin, csc, tan, cot SIGNS OF THE TRIGONOMETRIC FUNCTIONS

  11. Fundamental Identities ● Reciprocal Identities: csct = 1/sin t sec t = 1/cost cot t = 1/tan t tan t = sin t/cost cot t = cost/sin t ● Pythagorean Identities: sin^2t + cos^t = 1 tan^2t + 1 = sec^2t 1 + cot^2t = csc^2t

  12. Trigonometric Graphs • Periodic Properties: The functions tan and cot have period π tan(x + π) = tan x cot(x + π) = cot x The functions csc and sec have period 2 π csc(x + 2π) = csc x sec(x + 2π)= sec x

  13. Period Properties of Sine and Cosine • The functions of Sine and Cosine both have a period of 2π • This means they repeat themselves after one full rotation around the unit circle

  14. The sine function starts from the origin • It then follows the pattern of Peak, Root, Valley • The roots are at every 1 Pi when the period is 2 Pi • The peaks are equal to the amplitude which is equal to the coefficient of the function. • Valleys are also derived from the amplitude The Function of Sine F(x)=sinx

  15. The Function of Cosine • The Cosine Function starts at a peak which is equal to • amplitude or coefficient of the function. • It then follows the pattern root, valley, peak. The roots • occurring at every 1/2Pi. • The valleys and peaks equal to the amplitude.

  16. Horizontal and Vertical Stretching of Cosine and Sine • Horizontal stretching occurs when you a have a change of • the period of the function. • Ex 1. sin2x would repeat itself twice in the one rotation of the unit circle. • Ex2. sin1/2x would repeat itself once in 2 rotations of the unit circle. • Vertical stretching occurs from a change in amplitude or the coefficient of • function. • Ex 1. 2sinx would have a peak and valley at 2 and -2 respectively.

  17. Function Shifts Horizontal shifts of the sine and cosine functions are shown as sin(x+a) where is some value in radians. Vertical shifts look like sinx+a which would move it up or down depending on (a).

  18. The Tangent Function • The tangent function has a period of Pi but starts out at negative ½ Pi • and goes to positive ½ Pi. • Its shape liked an “s” and intersects the origin in the middle • It also has asymptotes' at the beginning and end of each period

  19. Reciprocal Functions • Cotangent = 1/tan : the reciprocal of tangent starts at the origin with an • asymptotes at the origin and has a period of 1 Pi where it ends with another • asymptote. It too looks like an “s” but it has a negative slope as it moves from • Positive infinity to negative infinity in its “Y” values. • Cosecant =1/sin : the reciprocal of sine has asymptotes at every ½ Pi . • If you take the peaks of the cosine function that is the vertex of the • Parabola formed by the reciprocal • Secant= 1/cos: the reciprocal of the cosine function is related to the • Cosecant function in that its parent function’s peaks are the vertices of the • Parabolas formed. However secant has asymptotes at 0 and 1 Pi instead • Of every ½ pi.

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