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

2.6 Rational Functions

A rational function can be written as






where N(x) and D(x) are polynomials and D(x) is not the zero polynomial.



Domain of a rational function of x is all real numbers except x-values that make the denominator zero.


Ex.




Domain:




Horizontal and Vertical Asymptotes

The line y = b is a horizontal asymptote of the graph of f if as or .

Ex 1

The graph of

has the line y = -1 as a horizontal asymptote. The degree of the of the numerator is equal to the degree of the denominator, and the horizontal asymptote is given by the ratio of the leading coefficents of the numerator and denominator.

Ex 2

The graph of

has no horizontal asymptote because the degree of the numerator is greater than the degree of the denominator.

Ex 3

The graph of

has the line y=0 (the x-axis) as a horizontal asymptote. Note that the degree of the numerator is less than the degree of the denominator.

The line x = a is a vertical asymptote of the graph of f if or


as , either from the left or right.


Ex.

The graph of

has a vertical asymptote of 3 because that is where D(x) = 0.



Intercepts


Y-intercept :





X-Intercept(s):

2.4: Complex Numbers

The Complex Number System:

i is an imaginary number

i = √-1, so i² = -1

standard form of a complex number: a + bi

Operations with Complex Numbers:

Addition: add like terms

(3 + 7i) + (9 + 6i)

3 + 9 + 7i + 6i

12 + 13i

Subtraction: subtract like terms

(4 + 5i) - (2-3i)

4-2 + 5i-(-3i)

2 + 8i

Multiplication: distribute

(3 + 2i)(1 + 2i)

3 + 6i + 2i + 2i²

3 + 8i + 2(-1)

1 + 8i

Division: multiply by the complex conjugate

Finding i to any power:

this pattern repeats itself forever and ever

So if you wanted to find i to a really high power, like:

you would need to divide the exponent, 327, by 4 to find what number in the pattern it stops at.

327/4 = 81 with a remainder of 3 (which is really the only part that matters to us)

Since the remainder is 3, the answer will be the same as i to the power of 3, which is -i

so: