Trigonometry Examples

$\tan (x) = \frac{5}{12}$ , $\cos (- 270 + x)$

Step 1

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

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

Step 2

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

$Hypotenuse = \sqrt{{opposite}^{2} + {adjacent}^{2}}$

Step 3

Replace the known values in the equation.

$Hypotenuse = \sqrt{{(5)}^{2} + {(12)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

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

Hypotenuse $= \sqrt{25 + {(12)}^{2}}$

Step 4.2

Raise $12$ to the power of $2$ .

Hypotenuse $= \sqrt{25 + 144}$

Step 4.3

Add $25$ and $144$ .

Hypotenuse $= \sqrt{169}$

Step 4.4

Rewrite $169$ as $13^{2}$ .

Hypotenuse $= \sqrt{13^{2}}$

Step 4.5

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

Hypotenuse $= 13$

Step 5

Use the definition of cosine to find the value of $\cos (x)$ .

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

Step 6

Substitute in the known values.

$\cos (x) = \frac{12}{13}$