Solve the trigonometric equation
[tex]\cos (\frac{x}{2})-\sin (x)=0[/tex]
To express both trigonometric functions in terms of the same argument, we have to use some identities.
Please note the cos is of x/2 and the sine is of x
We'll use the identity known as the sine of the double angle:
sin(2A)=2sin(A)cos(A)
in this case, A=x/2, and the identity is now:
sin(x)=2sin(x/2)cos(x/2)
note we have now all the trigonometric functions with the same argument x/2
Substituting the new-found relationship in the original equation, we have
[tex]\cos (\frac{x}{2})-2\sin (\frac{x}{2})\cos (\frac{x}{2})=0[/tex]
Now, we factor the expression by cos(x/2):
[tex]\cos (\frac{x}{2})(1-2\sin (\frac{x}{2}))=0[/tex]
We have now a factored equation, both factors can be zero, which gives us two possible solutions:
[tex]\cos (\frac{x}{2})=0,or1-2\sin (\frac{x}{2})=0[/tex]
The first solution comes from cos(x/2)=0
There are several angles whose cosine is 0. They are 90°, 270°, 450°, and many more, thus:
x/2=90°
Solving
x=180°
This is our first solution since the angle lies in the interval (0,360°)
Now for the second solution:
[tex]1-2\sin (\frac{x}{2})=0[/tex]
Solving for sin(x/2):
[tex]\sin (\frac{x}{2})=\frac{1}{2}[/tex]
There are many angles whose sine is 1/2, like 30°, 150°, and others. Taking the first angle:
x/2=30° or x=60° is our second solution. One more remains
x/2=150° or x=300°
Summarizing, the whole set of solutions is
60°, 180°, 300°
The third choice is correct
