Precalculus Examples

\cos (θ) = - \frac{24}{25}

$\cos (θ) = - \frac{24}{25}$

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 (θ) = \frac{adjacent}{hypotenuse}

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

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}}

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

Step 3

Replace the known values in the equation.

Opposite = - \sqrt{{(25)}^{2} - {(- 24)}^{2}}

$Opposite = - \sqrt{{(25)}^{2} - {(- 24)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Negate

\sqrt{{(25)}^{2} - {(- 24)}^{2}}

$\sqrt{{(25)}^{2} - {(- 24)}^{2}}$ .

Opposite

= - \sqrt{{(25)}^{2} - {(- 24)}^{2}}

$= - \sqrt{{(25)}^{2} - {(- 24)}^{2}}$

Step 4.2

Raise

25

$25$ to the power of

2

$2$ .

Opposite

= - \sqrt{625 - {(- 24)}^{2}}

$= - \sqrt{625 - {(- 24)}^{2}}$

Step 4.3

Raise

- 24

$- 24$ to the power of

2

$2$ .

Opposite

= - \sqrt{625 - 1 \cdot 576}

$= - \sqrt{625 - 1 \cdot 576}$

Step 4.4

Multiply

- 1

$- 1$ by

576

$576$ .

Opposite

= - \sqrt{625 - 576}

$= - \sqrt{625 - 576}$

Step 4.5

Subtract

576

$576$ from

625

$625$ .

Opposite

= - \sqrt{49}

$= - \sqrt{49}$

Step 4.6

Rewrite

49

$49$ as

7^{2}

$7^{2}$ .

Opposite

= - \sqrt{7^{2}}

$= - \sqrt{7^{2}}$

Step 4.7

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

Opposite

= - 1 \cdot 7

$= - 1 \cdot 7$

Step 4.8

Multiply

- 1

$- 1$ by

7

$7$ .

Opposite

= - 7

$= - 7$

Opposite

= - 7

$= - 7$

Step 5

Find the value of sine.

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Step 5.1

Use the definition of sine to find the value of

\sin (θ)

$\sin (θ)$ .

\sin (θ) = \frac{opp}{hyp}

$\sin (θ) = \frac{opp}{hyp}$

Step 5.2

Substitute in the known values.

\sin (θ) = \frac{- 7}{25}

$\sin (θ) = \frac{- 7}{25}$

Step 5.3

Move the negative in front of the fraction.

\sin (θ) = - \frac{7}{25}

$\sin (θ) = - \frac{7}{25}$

\sin (θ) = - \frac{7}{25}

$\sin (θ) = - \frac{7}{25}$

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 (θ)

$\tan (θ)$ .

\tan (θ) = \frac{opp}{adj}

$\tan (θ) = \frac{opp}{adj}$

Step 6.2

Substitute in the known values.

\tan (θ) = \frac{- 7}{- 24}

$\tan (θ) = \frac{- 7}{- 24}$

Step 6.3

Dividing two negative values results in a positive value.

\tan (θ) = \frac{7}{24}

$\tan (θ) = \frac{7}{24}$

\tan (θ) = \frac{7}{24}

$\tan (θ) = \frac{7}{24}$

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 (θ)

$\cot (θ)$ .

\cot (θ) = \frac{adj}{opp}

$\cot (θ) = \frac{adj}{opp}$

Step 7.2

Substitute in the known values.

\cot (θ) = \frac{- 24}{- 7}

$\cot (θ) = \frac{- 24}{- 7}$

Step 7.3

Dividing two negative values results in a positive value.

\cot (θ) = \frac{24}{7}

$\cot (θ) = \frac{24}{7}$

\cot (θ) = \frac{24}{7}

$\cot (θ) = \frac{24}{7}$

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 (θ)

$\sec (θ)$ .

\sec (θ) = \frac{hyp}{adj}

$\sec (θ) = \frac{hyp}{adj}$

Step 8.2

Substitute in the known values.

\sec (θ) = \frac{25}{- 24}

$\sec (θ) = \frac{25}{- 24}$

Step 8.3

Move the negative in front of the fraction.

\sec (θ) = - \frac{25}{24}

$\sec (θ) = - \frac{25}{24}$

\sec (θ) = - \frac{25}{24}

$\sec (θ) = - \frac{25}{24}$

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 (θ)

$\csc (θ)$ .

\csc (θ) = \frac{hyp}{opp}

$\csc (θ) = \frac{hyp}{opp}$

Step 9.2

Substitute in the known values.

\csc (θ) = \frac{25}{- 7}

$\csc (θ) = \frac{25}{- 7}$

Step 9.3

Move the negative in front of the fraction.

\csc (θ) = - \frac{25}{7}

$\csc (θ) = - \frac{25}{7}$

\csc (θ) = - \frac{25}{7}

$\csc (θ) = - \frac{25}{7}$

Step 10

This is the solution to each trig value.

\sin (θ) = - \frac{7}{25}

$\sin (θ) = - \frac{7}{25}$

\cos (θ) = - \frac{24}{25}

$\cos (θ) = - \frac{24}{25}$

\tan (θ) = \frac{7}{24}

$\tan (θ) = \frac{7}{24}$

\cot (θ) = \frac{24}{7}

$\cot (θ) = \frac{24}{7}$

\sec (θ) = - \frac{25}{24}

$\sec (θ) = - \frac{25}{24}$

\csc (θ) = - \frac{25}{7}

$\csc (θ) = - \frac{25}{7}$