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

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

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

 See example  Use equation: x2 + y2 = 1

Terminal Points Terminal Point – the point P(x,y) obtained and determined by the real number t 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 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 See example t = -π

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

To find the terminal point P determined by any value of t, use the following steps… 1. Find the reference number t 2. Find the terminal point Q(a, b) determined by t 3. 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

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

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 Even Odd

SIGNS OF THE TRIGONOMETRIC FUNCTIONS QuadrantPositive FunctionsNegative functions Iall none II sin, csccos, sec, tan, cot III tan, cotsin, csc, cos, sec IV cos, secsin, csc, tan, cot

Fundamental Identities ● Reciprocal Identities: csc t = 1/sin tsec t = 1/cos t cot t = 1/tan ttan t = sin t/cos t cot t = cos t/sin t ● Pythagorean Identities: sin^2t + cos^t = 1 tan^2t + 1 = sec^2t 1 + cot^2t = csc^2t

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

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

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 F(x)=sinx

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

 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.

 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).

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

 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.