Question

1. On the interval [0, 2pi] what are the solutions to the equation sin3xcos2x =
-cos3xsin2x + 1?
pi/10 and pi/2?

2. What is the value of tan75degrees?
√(3) + 1)/(1 - √(3))?

3. Value of cos(130degrees)cos(130degrees) + sin(10degrees)sin(10degrees)?
Not sure

4. On the interval [0, 2pi] what are the solutions to sin2xcos3x = cos2xsin3x - 1/2
pi/6 and 5pi/6?

5. Value of sin(4pi/9)cos(5pi/18) - cos(4pi/9)sin(5pi/18)?
I'm not sure about my answers, is it root 3/2?

Answer 1

#1. Since
sin3xcos2x + cos3xsin2x = sin5x you have
sin5x = 1
So, 5x = π/2, 5π/2, ...
and x = π/10, 5π/10, 9π/10, 13π/10, 17π/10

You have to keep adding 2π/5 until x gets to 2π

#2. correct

#3. Since cos(a-b) = cosa cosb - sina sinb you have
cos120° = -cos60° = -1/2

#4. Work like #1
sin5x = 1/2

#5. sin (4/9 - 5/18)pi = sin(pi/6) = 1/2
#4.