Amy's Reflection #36

Since our trig exam is coming up here a review on chapter 10...

Sum and Difference formulas for Cosine and Sine:

cos (alpha + or - beta) = cos(alpha)cos(beta) - or + sin(alpha)sin(beta)
sin (alpha + or - beta) = sin(alpha)cos(beta) + or - cos(alpha)sin(beta)

Rewriting a Sum or Difference as a Product:

sin(x) + sin(y) = 2sin(x+y/2)cos(x-y/2)
sin(x) - sin(y) = 2cos(x+y/2)sin(x-y/2)
cos(x) + cos(y) = 2cos(x+y/2)cos(x-y/2)
cos(x) - cos(y) = -2sin(x+y/2)sin(x-y/2)
**we didn't use these formulas for anything..so i got no idea where to use them..

Half-Angle and Double Angle Formulas:

sin(2alpha) = 2sin(alpha)cos(alpha)
cos(2alpha) = cos^2(alpha)-sin^2(alpha)=1-2sin^2(alpha)=2cos^2
(alpha)-1
tan(2alpha) = 2tan(alpha)/1-tan^2(alpha)
sin(alpha/2)= +- sqrt(1-cos(alpha)/2)
cos(alpha/2)= +- sqrt(1+cos(alpha)/2)
tan(alpha/2)= +- sqrt(1-cos(alpha)/1+cos(alpha))=sin(alpha)/1+cos
(alpha)=1-cos(alpha)/sin(alpha)

now here are some examples:

1. tan α = 2 and tan β=1, find tan (α - β)

= tan α + tan β/1-tan α tan β

=2+1/1-(2)(6)

=3/-1

=-3

2. Find the exact value of: tan 15+tan 30/1-tan 15 tan 30

tan α = 2 and tan β=1

find tan (α - β)

= tan (15 + 30)

=tan (45)

=1

3. Find the exact value of sin 15degrees

*exact value means you use your trig chart
*think of two numbers from the trig chart can either add or subtract to give you 15
*since it's (45-30), you would look for the formula that uses sin

sin (a-B) = sin a cos B - cos a sin B

* plug #s into equation..

a=45 degrees B=30 degrees

sin (45-30) = sin 45 cos 30 - cos 45 sin 30

sin (a-B) = sin a cos B - cos a sin B

sin (45-30) = sin 45 cos 30 - cos 45 sin 30

sin 15 = (square root of 2 over 2)(square root of 3 over 2) - (square root of 2 over 2)(1/2)

sin 15 degrees = (square root of 6 over 4) - (square root of 2 over 4)

= square root of 6 - square root of 2 all over 4

i hope this will help refreshen someone's memory :)