Question

11. Find the exact value by using a half-angle identity. (1 point)
sin 22.5°


a) negative one half times the square root of quantity two plus square root of two.
b) one half times the square root of quantity two plus square root of two.
c) negative one half times the square root of quantity two minus square root of two.
d) one half times the square root of quantity two minus square root of two

12. Find all solutions to the equation in the interval [0, 2π). (1 point)
cos x = sin 2x


a) pi divided by two. , three pi divided by two.
b)pi divided by six. , pi divided by two. , five pi divided by six. , three pi divided by two.
c) 0, π
d) 0, pi divided by six. , five pi divided by six. , π

13. Rewrite with only sin x and cos x. (1 point)
sin 2x - cos x


a) 2 sin x cos2x
b) sin x
c) cos x (2 sin x - 1)
d) 2 sin x

14. Verify the identity. (1 point)
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.

15. Verify the identity. (1 point)
cot x minus pi divided by two. = -tan x



PLEASE HELP

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