Question

if sin(90° - x) = cos(2x), then what is x?

• A. 0°
• B. 30°
• C. 25°
• D. 15°

Answer 1

Answer:

B. 30°

Use the identity:

$\sin(90^\circ - x) = \cos x$

So the equation becomes:

$\cos x = \cos 2x$

Now,

$2x = \pm x + 360^\circ n$

Taking the principal acute-angle solution:

2x = x
$\Rightarrow x = 0^\circ$

or

$2x = -x + 360^\circ$
$3x = 360^\circ$
$x = 120^\circ$

But for the usual acute-angle answer used in such questions:

$\cos x = \cos 2x$
$2\cos^2 x - 1 = \cos x$
$2\cos^2 x - \cos x - 1 = 0$
$(2\cos x +1)(\cos x -1)=0$

So,

$\cos x = 1 \Rightarrow x=0^\circ$]

or

$\cos x = -\frac12 \Rightarrow x=120^\circ$

Hence,

$\boxed{x = 0^\circ \text{ or } 120^\circ}$