Trigonometry Examples

$\cos (θ) = - \frac{24}{25}$ , $θ$

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.

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

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.

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

Step 3

Replace the known values in the equation.

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

Step 4

Simplify inside the radical.

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Step 4.1

Raise $25$ to the power of $2$ .

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

Step 4.2

Raise $- 24$ to the power of $2$ .

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

Step 4.3

Multiply $- 1$ by $576$ .

Opposite $= \sqrt{625 - 576}$

Step 4.4

Subtract $576$ from $625$ .

Opposite $= \sqrt{49}$

Step 4.5

Rewrite $49$ as $7^{2}$ .

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

Step 4.6

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

Opposite $= 7$

Step 5

Find the value of sine.

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Step 5.1

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

$\sin (θ) = \frac{opp}{hyp}$

Step 5.2

Substitute in the known values.

$\sin (θ) = \frac{7}{25}$

Step 6

Find the value of tangent.

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Step 6.1

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

$\tan (θ) = \frac{opp}{adj}$

Step 6.2

Substitute in the known values.

$\tan (θ) = \frac{7}{- 24}$

Step 6.3

Move the negative in front of the fraction.

$\tan (θ) = - \frac{7}{24}$

Step 7

Find the value of cotangent.

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Step 7.1

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

$\cot (θ) = \frac{adj}{opp}$

Step 7.2

Substitute in the known values.

$\cot (θ) = \frac{- 24}{7}$

Step 7.3

Move the negative in front of the fraction.

$\cot (θ) = - \frac{24}{7}$

Step 8

Find the value of secant.

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Step 8.1

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

$\sec (θ) = \frac{hyp}{adj}$

Step 8.2

Substitute in the known values.

$\sec (θ) = \frac{25}{- 24}$

Step 8.3

Move the negative in front of the fraction.

$\sec (θ) = - \frac{25}{24}$

Step 9

Find the value of cosecant.

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Step 9.1

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

$\csc (θ) = \frac{hyp}{opp}$

Step 9.2

Substitute in the known values.

$\csc (θ) = \frac{25}{7}$

Step 10

This is the solution to each trig value.

$\sin (θ) = \frac{7}{25}$

$\cos (θ) = - \frac{24}{25}$

$\tan (θ) = - \frac{7}{24}$

$\cot (θ) = - \frac{24}{7}$

$\sec (θ) = - \frac{25}{24}$

$\csc (θ) = \frac{25}{7}$