Precalculus Examples

$\tan (x) = - \frac{15}{8}$

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.

$Hypotenuse = \sqrt{{(15)}^{2} + {(- 8)}^{2}}$

Step 4

Simplify inside the radical.

Tap for more steps...

Step 4.1

Raise $15$ to the power of $2$ .

Hypotenuse $= \sqrt{225 + {(- 8)}^{2}}$

Step 4.2

Raise $- 8$ to the power of $2$ .

Hypotenuse $= \sqrt{225 + 64}$

Step 4.3

Add $225$ and $64$ .

Hypotenuse $= \sqrt{289}$

Step 4.4

Rewrite $289$ as $17^{2}$ .

Hypotenuse $= \sqrt{17^{2}}$

Step 4.5

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

Hypotenuse $= 17$

Step 5

Find the value of sine.

Tap for more steps...

Step 5.1

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

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

Step 5.2

Substitute in the known values.

$\sin (x) = \frac{15}{17}$

Step 6

Find the value of cosine.

Tap for more steps...

Step 6.1

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

$\cos (x) = \frac{adj}{hyp}$

Step 6.2

Substitute in the known values.

$\cos (x) = \frac{- 8}{17}$

Step 6.3

Move the negative in front of the fraction.

$\cos (x) = - \frac{8}{17}$

Step 7

Find the value of cotangent.

Tap for more steps...

Step 7.1

Use the definition of cotangent to find the value of $\cot (x)$ .

$\cot (x) = \frac{adj}{opp}$

Step 7.2

Substitute in the known values.

$\cot (x) = \frac{- 8}{15}$

Step 7.3

Move the negative in front of the fraction.

$\cot (x) = - \frac{8}{15}$

Step 8

Find the value of secant.

Tap for more steps...

Step 8.1

Use the definition of secant to find the value of $\sec (x)$ .

$\sec (x) = \frac{hyp}{adj}$

Step 8.2

Substitute in the known values.

$\sec (x) = \frac{17}{- 8}$

Step 8.3

Move the negative in front of the fraction.

$\sec (x) = - \frac{17}{8}$

Step 9

Find the value of cosecant.

Tap for more steps...

Step 9.1

Use the definition of cosecant to find the value of $\csc (x)$ .

$\csc (x) = \frac{hyp}{opp}$

Step 9.2

Substitute in the known values.

$\csc (x) = \frac{17}{15}$

Step 10

This is the solution to each trig value.

$\sin (x) = \frac{15}{17}$

$\cos (x) = - \frac{8}{17}$

$\tan (x) = - \frac{15}{8}$

$\cot (x) = - \frac{8}{15}$

$\sec (x) = - \frac{17}{8}$

$\csc (x) = \frac{17}{15}$