Amy's Reflection #37

so we pretty much just reviewed for the test..here's some examples of the stuff we gonna have to know..

Example 1: Find the exact value of sin 80 cos 130+ cos 80 sin 130

Let alpha = 80 and beta = 130 then sin 80 cos 130 + cos 80sin 130 = sin alpha cos beta + cos alpha sin b

= sin (alpha + beta)
= sin (80+130)
= sin (210)
= sin (180 + 30)
= - sin 30
= -1/2

Example 2: Find the exact value of

cos^215 - sin^215 = cos(2alpha)
= cos(30)
= sqrt 3/2

Example 3: cos 15

alpha = 45, beta = 30

cos (alpha - beta) = cos 45 cos30 + sin45 sin 30

cos (45 - 30) = (√(2/2) (√(3/2) + (√(2/2) (1/2)

= √(√6 + √2)/all over 4

Example 4: (1 + cot^2) (1 - cos 2x)

(csc^2x) (1-cos2x)

(csc^2x) (1-(1 - 2sin^2x)

(1/sin^2x) (2sin^2x)

= 2

Example 4: Find the exact value of sin22.5

alpha=2(22.5)

alpha=45

**if you are given an angle with a decimal you use the half-angle formula. To find alpha, you multiply by two.

Example 5: Sum Formula for Cosine

cos 75 cos 15 + sin 75 sin 15

=cos (75-15) = cos 60 = 1/2

Example 6: Find the exact value of tan15 +tan30/1-tan15 (tan30)

= tan(15 + 30)

= tan(45)

= 1

Examples:

1. Express 2 cis 50degrees in rectangular form

2 cos 50 + 2 sin 50 i

2. Express -1-2i in polar form

radius = +- sqrt of ((-1)^2 + (-2)^2)) = +- sqrt of (5)

theta = tan^-1(-2/-1)

theta = tan^-1(1)

*tangent is positive in the first and third quadrants, 63.435 and 243.435
*63 is positive for cosine so it goes with the positive sqrt of 5
*243 is negative for cosine so it goes with the negative sqrt of 5

z= sqrt of 5 cis 63.435

z= sqrt of 5 cos 63.435 + sqrt of 5 sin 63.435 i

z= negative sqrt of 5 cis 243.435

z= negative sqrt of 5 cos 243.435 + negative sqrt of 5 sin 243.435 i

De Moivre's Theorem: z^n = r^n cis(n)(theta)

Examples:

1. z=2cis20degrees Find z^2

z^2=2^2cis2(20degrees)

z^2=4cis40degrees

2. 4cis15degrees Find z^4

z^4=4^4cis4(15degrees)

z^4=256cis60degrees

Limacon
r=a+b sin(theta)
r=a+b cos(theta)

Cardioid
a-b sin(theta)
r=a-b cos(theta)

Rose
r=a sin(n theta)
r=a cos (n theta)

*n=how many petals

Archimedes Spiral
r=a theta+b

Logarithmic Spiral
r=a b^theta

Examples:
1. r=theta+2
2. r=2+3cos(theta)
3. r=5
4. r=3sin(4 theta)
5. r=1/2(3^theta)
6. r=2sin(theta)

1. Archimedes spiral
2. limacon
3. circle with its center at the pole
4. rose with 4 petals
5. logarithmic spiral
6. circle that intersects with the pole

i hope this helped..don't forget to study for our trig!!!