Graphs of Csc, Sec, Cot and Tan

a = Vertical stretch or compress

b = Horizontal stretch or compress

c = Shifts left or right

d = Shift up or down

CSC Graph

y = csc(x)

SEC Graph

y= sec(x)

TAN Graph

y= tan(x)

COT Graph

y= cot(x)

4.5 Graphs of Sine and Cosine Functions

or

The constant factor of a in y=asinx acts as a scaling factor- a vertical stretch or vertical shrink of the basic sine curve. If , the basic sine curve is stretched, and if , the basic sine curve is shrunk. The result is that the graph of y=asinx ranges between -a and a instead of -1 and 1.

The amplitude of and represents half the distance between the maximum and minimum values of the function and is given by:

Let b be a positive real number. The period of and is given by:

Note that if , the period of is greater than and represents a horizontal stretching of the graph of . Similarly, if , the period of is less than and represents a horizontal shrinking of the graph of .

The constant c creates horizontal translations (shifts) of the basic sine and cosine curves.

The graph of is shifted by an amount of . The number is the phase shift.

The left and right endpoints of a one-cycle interval can be determined by solving the equations and .

4.4 Reference Angles

Reference Angles - the corresponding acute angles to angles greater then (or less then ). It is formed by the terminal side of and the horizontal axis. It is written in terms of theta prime ().

In the 1st quadrant the reference angle is the angle itself. ()

In the 2nd quadrant the reference angle is got by subtracting the given angle by . ()

In the 3rd quadrant the reference angle is got by subtracting from the given angle. ()

In the 4th quadrant the reference angle is got by subtracting the given angle by . ()

4.3 Identities

Identities are equations that are always true.

5 + 2(x-3) = 2x - 1

5 + 2x-6 = 2x - 1

5 - 6 = -1

-1 = -1

Reciprocal Identities

sin(theta) = 1/csc(theta) csc(theta) = 1/sin(theta)

cos(theta) = 1/sec(theta) sec(theta) = 1/cos(theta)

tan(theta) = 1/cot(theta) cot(theta) = 1/tan(theta)

Quotient Identities

tan(theta) = sin(theta)/cos(theta) cot(theta) = cos(theta)/sin(theta)

Even/Odd Identities

sin(-theta) = -sin(theta) csc(-theta) = -csc(theta)

cos(-theta) = cos(theta) sec(-theta) = sec(theta) - (only even functions)

tan(-theta) = -tan(theta) cot(-theta) = -cot(theta)

Pythagorean Identities

a^2 + b^2 = c^2

(c x sinA)^2 + (c x cosA)^2 = c^2

c^2(sinA)^2

C^2 x sin^2A + c^2 x cos^2A= C^2

sin^2A + cos^2A = 1

sin^2(theta) + cos^2(theta) = 1

tan^2(theta) + 1 = sec^2(theta)

cot^2(theta) + 1 = csc^2(theta)

4.2 Trigonometric Functions: The Unit Circle

Definitions of Trigonometric Functions

Let t be a real number and let (x,y) be the point on the unit circle corresponding to t.

sin t = y csc t = 1/y y cannot =0

cos t = x sec t = 1/x x cannot =0

tan t = y/x cot t = x/y y cannot =0

In Other Words

SOHCAHTOA CHOSHACAO

sin = opp/hyp csc = hyp/opp

cos = adj/hyp sec = hyp/adj

tan =opp/adj cot = adj/hyp

Even and Odd Trigonometric Functions

The cosine and secant functions are even

cos(-t) = cos(t) sec(-t) = sec t

The sine, cosecant, tangent, and cotangent functions are odd.

sin(-t) = -sin(t) csc(-t) = sec(t)

tan(-t) = -tan(t) cot(-t) = -cot(t)

4.1

You can measure angles in radians.

One Radian is the measure of a central angle that intercepts an arc s equal in length to the radius r of the circle.

Complementary angles are angles that add up to 90 degrees.

Supplementary angles are angles that add up to 180 degrees.

To convert degrees to radians, multiply degrees by (pie)rad/180.

To convert radians to degrees, multiply radians by 180/(pie)rad.

Graphing Ration Functions:

-To graph a rational function, you find the asymptotes and the intercepts

Example:

X-intercept: (set the numerator equal to 0) 2x=0; 0

Y-intercept: ( plug 0 in for x in the equation) 0

Vertical asymptote: (Denominator) Horizontal asymptote: (exponents)

y=0

(x+2)(x-1)

x=-2, x=1

-To find out which way the line goes, plug in an x value thats appropriate for you asymptote and that will tell you if its positive or negative

-On Occasion there can be a hole, which means there is a multiplicity