Precalculus Examples

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

Step 1

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

$\sin (x) = \frac{opposite}{hypotenuse}$

Step 2

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

$Adjacent = - \sqrt{{hypotenuse}^{2} - {opposite}^{2}}$

Step 3

Replace the known values in the equation.

$Adjacent = - \sqrt{{(13)}^{2} - {(12)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Negate $\sqrt{{(13)}^{2} - {(12)}^{2}}$ .

Adjacent $= - \sqrt{{(13)}^{2} - {(12)}^{2}}$

Step 4.2

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

Adjacent $= - \sqrt{169 - {(12)}^{2}}$

Step 4.3

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

Adjacent $= - \sqrt{169 - 1 \cdot 144}$

Step 4.4

Multiply $- 1$ by $144$ .

Adjacent $= - \sqrt{169 - 144}$

Step 4.5

Subtract $144$ from $169$ .

Adjacent $= - \sqrt{25}$

Step 4.6

Rewrite $25$ as $5^{2}$ .

Adjacent $= - \sqrt{5^{2}}$

Step 4.7

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

Adjacent $= - 1 \cdot 5$

Step 4.8

Multiply $- 1$ by $5$ .

Adjacent $= - 5$

Step 5

Find the value of cosine.

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Step 5.1

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

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

Step 5.2

Substitute in the known values.

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

Step 5.3

Move the negative in front of the fraction.

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

Step 6

Find the value of tangent.

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Step 6.1

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

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

Step 6.2

Substitute in the known values.

$\tan (x) = \frac{12}{- 5}$

Step 6.3

Move the negative in front of the fraction.

$\tan (x) = - \frac{12}{5}$

Step 7

Find the value of cotangent.

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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.

$\cot (x) = \frac{- 5}{12}$

Step 7.3

Move the negative in front of the fraction.

$\cot (x) = - \frac{5}{12}$

Step 8

Find the value of secant.

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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.

$\sec (x) = \frac{13}{- 5}$

Step 8.3

Move the negative in front of the fraction.

$\sec (x) = - \frac{13}{5}$

Step 9

Find the value of cosecant.

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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.

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

Step 10

This is the solution to each trig value.

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

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

$\tan (x) = - \frac{12}{5}$

$\cot (x) = - \frac{5}{12}$

$\sec (x) = - \frac{13}{5}$

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