Addition and Subtraction Identities Sum to Product and Product to Sum Identities

Question

1. Write the product as a sum or difference:

10sin(44q)sin(10q)=

2. Rewrite sin(x−5π6) in terms of sin(x) and cos(x)

3. Let f(x)=−2.4sin(x)−5.3cos(x). What is the maximum and minimum value of this function?

4. Solve −sin(x)+5cos(x)=4 for the first 2 positive solutions

5. Solve sin(2x)cos(4x)−cos(2x)sin(4x)=−0.25 for the smallest positive solution.

Answer 1

1

10sin44qsin10q

=(10)(1/2)(cos(42q-10q)-cos(42q+10q)

=5(cos32q-cos54q)

=5cos32q -5cos54q

2

sin(x-5pi/6)

=sinxcos5pi/6 -cosxsin5pi/6

=-(sqr3/2)sinx-(1/2)cosx

=-(1/2)(sqr3sinx +cosx)

3

f(x)= -2.4sinx -5.3cosx

f'(x)= -2.4cosx +5.3sinx =0

tanx = 2.4/5.3

x= tan^-1(2.4/5.3)

x= about 24.36 +180n degrees

where n=any integer

are minimums when n is even, max when odd

4

-sinx +5cosx = 4

sinx+4 = 5cosx

sinx = 5sqr(1-sin^2(x))

sin^2(x)=25-25sin^2(x)

sin^2(x)=25/26

sinx =+/-5/sqr26

x = sin^-1(+/-5/sqr26)

least 2 positives are

about 63.9

and 423.9 degrees

5

sin2xcos4x- cos2xsin4x =-0.25

use sum & difference formula

sin(2x-4x)=-1/4

sin(-2x)=-1/4

odd function

-sin2x =-1/4

multiply by -1

sin2x=1/4

2x=sin^-1(1/4)

x=(1/2)sin^-1(1/4)

x=about 7.24 degrees