Find the Other Trig Values in Quadrant I cos(2x)=5/13

$\cos (2 x) = \frac{5}{13}$

Use the definition of cosine to find the known sides of the unit circle right triangle. The quadrant determines the sign on each of the values.

$\cos (2 x) = \frac{adjacent}{hypotenuse}$

Find the opposite side of the unit circle triangle. Since the adjacent side and hypotenuse are known, use the Pythagorean theorem to find the remaining side.

$Opposite = \sqrt{{hypotenuse}^{2} - {adjacent}^{2}}$

Replace the known values in the equation.

$Opposite = \sqrt{{(13)}^{2} - {(5)}^{2}}$

Simplify inside the radical.

Raise $13$ to the power of $2$ .

Opposite $= \sqrt{169 - {(5)}^{2}}$

Raise $5$ to the power of $2$ .

Opposite $= \sqrt{169 - 1 \cdot 25}$

Multiply $- 1$ by $25$ .

Opposite $= \sqrt{169 - 25}$

Subtract $25$ from $169$ .

Opposite $= \sqrt{144}$

Rewrite $144$ as $12^{2}$ .

Opposite $= \sqrt{12^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

Opposite $= 12$

Find the value of sine.

Use the definition of sine to find the value of $\sin (2 x)$ .

$\sin (2 x) = \frac{opp}{hyp}$