If sin⁡2x=513 and 0∘&lt;x&lt;45∘, find sin⁡x and cos⁡x.

Question

If sin⁡2x=513 and 0∘&amp;lt;x&amp;lt;45∘, find sin⁡x and cos⁡x.

louran20z47 · Accepted Answer

Suppose we had a right triangle with an angle

2 x

, and

\sin 2 x = \frac{5}{13}

. Further suppose that the hypotenuse of the triangle was 13. We can deduce that the side oppoites

2 x

must be 5. Applying the Pythagorean theorem to find the other side we have

13^{2} = 5^{2} + {(adjacent side)}^{2}

169 - 25 = {(adjacent side)}^{2}

144 = {(adjacent side)}^{2}

implying that the side opposite angle 2x is 12. This allows us to state that

\cos 2 x = \frac{12}{13}

Which is easier to work with because

\cos 2 x = \cos^{2} x - \sin^{2} x = 2 \cos^{2} x - 1

Substituting we have

\frac{12}{13} = 2 \cos^{2} x - 1

\cos^{2} x = \frac{25}{26}

\cos x = \frac{5}{\sqrt{26}} = \frac{5 \sqrt{26}}{26}

Applying the Pythagorean identity we have

\cos^{2} x + \sin^{2} x = 1

\frac{25}{26} + \sin^{2} x = 1

\sin^{2} x = \frac{1}{26}

\sin x = \frac{1}{\sqrt{26}} = \frac{\sqrt{26}}{26}

If \sin2x=\frac{5}{13} and 0^\circ<x<45^\circ, find \sin x and \cos x.