Question

11. Find the exact value by using a half-angle identity. (1 point)

Answer 1

11. Find the exact value by using a half-angle identity. 
sin (22.5)
the sine half-angle formula

sin(x/2)=±((1−cos(x))/2) ^0.5         cos 45=(2^0.5)/2

sin(22.5)=±((1−cos(45))/2) ^0.5

sin(22.5)=±((2-2^0.5))^0.5/2

sin(22.5)=±0.3826834324

12. Find all solutions to the equation in the interval [0, 2π)
cos x = sin 2x
cosx-sin 2x=0
using a graphical tool
in the interval [0, 2π)

the solutions are
x1=0----------------not solution
x2=π/6------------ not solution
x3=π/2------------ is a solution
x4=5π/6---------- not solution
x5=3π/2---------- is a solution

the answer is the letter a) pi divided by two. , three pi divided by two

13. Rewrite with only sin x and cos x. sin(2x) = 2*sin(x)*cos(x)

sin 2x - cos x=2*sin(x)*cos(x)- cos x= cos x*(2*sin(x)-1)

the answer is  the letter c) cos x (2 sin x - 1)

14. Verify the identity. 
cosine of x divided by quantity one plus sine of x plus quantity one plus sine of x divided by cosine of x equals two times secant of x.

cosx/(1+sinx) + (1+sinx)/cosx 

= (cosx * cosx + (1+sinx)(1+sinx)) / (cosx (1+sinx)) 
= (cos²x + sin²x + 2 sinx + 1) / (cosx (1+sinx)) 
= (1 + 2 sinx + 1) / (cosx (1+sinx)) 
= (2 + 2 sinx) / (cosx (1+sinx)) 
= 2 (1+sinx) / (cosx (1+sinx)) 
= 2/cosx 
= 2 secx  Ok is correct