Precalculus Examples

\sin (x) = - \frac{4}{5}

$\sin (x) = - \frac{4}{5}$

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}

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

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}}

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

Step 3

Replace the known values in the equation.

Adjacent = - \sqrt{{(5)}^{2} - {(- 4)}^{2}}

$Adjacent = - \sqrt{{(5)}^{2} - {(- 4)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Negate

\sqrt{{(5)}^{2} - {(- 4)}^{2}}

$\sqrt{{(5)}^{2} - {(- 4)}^{2}}$ .

Adjacent

= - \sqrt{{(5)}^{2} - {(- 4)}^{2}}

$= - \sqrt{{(5)}^{2} - {(- 4)}^{2}}$

Step 4.2

Raise

5

$5$ to the power of

2

$2$ .

Adjacent

= - \sqrt{25 - {(- 4)}^{2}}

$= - \sqrt{25 - {(- 4)}^{2}}$

Step 4.3

Raise

- 4

$- 4$ to the power of

2

$2$ .

Adjacent

= - \sqrt{25 - 1 \cdot 16}

$= - \sqrt{25 - 1 \cdot 16}$

Step 4.4

Multiply

- 1

$- 1$ by

16

$16$ .

Adjacent

= - \sqrt{25 - 16}

$= - \sqrt{25 - 16}$

Step 4.5

Subtract

16

$16$ from

25

$25$ .

Adjacent

= - \sqrt{9}

$= - \sqrt{9}$

Step 4.6

Rewrite

9

$9$ as

3^{2}

$3^{2}$ .

Adjacent

= - \sqrt{3^{2}}

$= - \sqrt{3^{2}}$

Step 4.7

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

Adjacent

= - 1 \cdot 3

$= - 1 \cdot 3$

Step 4.8

Multiply

- 1

$- 1$ by

3

$3$ .

Adjacent

= - 3

$= - 3$

Adjacent

= - 3

$= - 3$

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)

$\cos (x)$ .

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

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

Step 5.2

Substitute in the known values.

\cos (x) = \frac{- 3}{5}

$\cos (x) = \frac{- 3}{5}$

Step 5.3

Move the negative in front of the fraction.

\cos (x) = - \frac{3}{5}

$\cos (x) = - \frac{3}{5}$

\cos (x) = - \frac{3}{5}

$\cos (x) = - \frac{3}{5}$

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)

$\tan (x)$ .

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

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

Step 6.2

Substitute in the known values.

\tan (x) = \frac{- 4}{- 3}

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

Step 6.3

Dividing two negative values results in a positive value.

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

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

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

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

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)

$\cot (x)$ .

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

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

Step 7.2

Substitute in the known values.

\cot (x) = \frac{- 3}{- 4}

$\cot (x) = \frac{- 3}{- 4}$

Step 7.3

Dividing two negative values results in a positive value.

\cot (x) = \frac{3}{4}

$\cot (x) = \frac{3}{4}$

\cot (x) = \frac{3}{4}

$\cot (x) = \frac{3}{4}$

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)

$\sec (x)$ .

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

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

Step 8.2

Substitute in the known values.

\sec (x) = \frac{5}{- 3}

$\sec (x) = \frac{5}{- 3}$

Step 8.3

Move the negative in front of the fraction.

\sec (x) = - \frac{5}{3}

$\sec (x) = - \frac{5}{3}$

\sec (x) = - \frac{5}{3}

$\sec (x) = - \frac{5}{3}$

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)

$\csc (x)$ .

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

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

Step 9.2

Substitute in the known values.

\csc (x) = \frac{5}{- 4}

$\csc (x) = \frac{5}{- 4}$

Step 9.3

Move the negative in front of the fraction.

\csc (x) = - \frac{5}{4}

$\csc (x) = - \frac{5}{4}$

\csc (x) = - \frac{5}{4}

$\csc (x) = - \frac{5}{4}$

Step 10

This is the solution to each trig value.

\sin (x) = - \frac{4}{5}

$\sin (x) = - \frac{4}{5}$

\cos (x) = - \frac{3}{5}

$\cos (x) = - \frac{3}{5}$

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

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

\cot (x) = \frac{3}{4}

$\cot (x) = \frac{3}{4}$

\sec (x) = - \frac{5}{3}

$\sec (x) = - \frac{5}{3}$

\csc (x) = - \frac{5}{4}

$\csc (x) = - \frac{5}{4}$