Trigonometry Examples

$\cos (2 x) = \frac{3}{5}$ , $\tan (x)$

Step 1

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.

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

Step 2

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.

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

Step 3

Replace the known values in the equation.

$Opposite = \sqrt{{(5)}^{2} - {(3)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

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

Opposite $= \sqrt{25 - {(3)}^{2}}$

Step 4.2

Raise $3$ to the power of $2$ .

Opposite $= \sqrt{25 - 1 \cdot 9}$

Step 4.3

Multiply $- 1$ by $9$ .

Opposite $= \sqrt{25 - 9}$

Step 4.4

Subtract $9$ from $25$ .

Opposite $= \sqrt{16}$

Step 4.5

Rewrite $16$ as $4^{2}$ .

Opposite $= \sqrt{4^{2}}$

Step 4.6

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

Opposite $= 4$

Step 5

Use the definition of tangent to find the value of $\tan (2 x)$ .

$\tan (2 x) = \frac{opposite}{adjacent}$

Step 6

Substitute in the known values.

$\tan (2 x) = \frac{4}{3}$