Precalculus Examples

$\cos (x) = \frac{24}{26}$

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.

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

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.

$Opposite = - \sqrt{{(26)}^{2} - {(24)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Negate $\sqrt{{(26)}^{2} - {(24)}^{2}}$ .

Opposite $= - \sqrt{{(26)}^{2} - {(24)}^{2}}$

Step 4.2

Raise $26$ to the power of $2$ .

Opposite $= - \sqrt{676 - {(24)}^{2}}$

Step 4.3

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

Opposite $= - \sqrt{676 - 1 \cdot 576}$

Step 4.4

Multiply $- 1$ by $576$ .

Opposite $= - \sqrt{676 - 576}$

Step 4.5

Subtract $576$ from $676$ .

Opposite $= - \sqrt{100}$

Step 4.6

Rewrite $100$ as $10^{2}$ .

Opposite $= - \sqrt{10^{2}}$

Step 4.7

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

Opposite $= - 1 \cdot 10$

Step 4.8

Multiply $- 1$ by $10$ .

Opposite $= - 10$

Step 5

Find the value of sine.

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Step 5.1

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

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

Step 5.2

Substitute in the known values.

$\sin (x) = \frac{- 10}{26}$

Step 5.3

Simplify the value of $\sin (x)$ .

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Step 5.3.1

Cancel the common factor of $- 10$ and $26$ .

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Step 5.3.1.1

Factor $2$ out of $- 10$ .

$\sin (x) = \frac{2 (- 5)}{26}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $2$ out of $26$ .

$\sin (x) = \frac{2 \cdot - 5}{2 \cdot 13}$

Step 5.3.1.2.2

Cancel the common factor.

$\sin (x) = \frac{2 \cdot - 5}{2 \cdot 13}$

Step 5.3.1.2.3

Rewrite the expression.

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

Step 5.3.2

Move the negative in front of the fraction.

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

Step 6

Cancel the common factor of $24$ and $26$ .

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Step 6.1

Factor $2$ out of $24$ .

$\cos (x) = \frac{2 (12)}{26}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $2$ out of $26$ .

$\cos (x) = \frac{2 \cdot 12}{2 \cdot 13}$

Step 6.2.2

Cancel the common factor.

$\cos (x) = \frac{2 \cdot 12}{2 \cdot 13}$

Step 6.2.3

Rewrite the expression.

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

Step 7

Find the value of tangent.

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Step 7.1

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

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

Step 7.2

Substitute in the known values.

$\tan (x) = \frac{- 10}{24}$

Step 7.3

Simplify the value of $\tan (x)$ .

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Step 7.3.1

Cancel the common factor of $- 10$ and $24$ .

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Step 7.3.1.1

Factor $2$ out of $- 10$ .

$\tan (x) = \frac{2 (- 5)}{24}$

Step 7.3.1.2

Cancel the common factors.

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Step 7.3.1.2.1

Factor $2$ out of $24$ .

$\tan (x) = \frac{2 \cdot - 5}{2 \cdot 12}$

Step 7.3.1.2.2

Cancel the common factor.

$\tan (x) = \frac{2 \cdot - 5}{2 \cdot 12}$

Step 7.3.1.2.3

Rewrite the expression.

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

Step 7.3.2

Move the negative in front of the fraction.

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

Step 8

Find the value of cotangent.

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Step 8.1

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

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

Step 8.2

Substitute in the known values.

$\cot (x) = \frac{24}{- 10}$

Step 8.3

Simplify the value of $\cot (x)$ .

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Step 8.3.1

Cancel the common factor of $24$ and $- 10$ .

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Step 8.3.1.1

Factor $2$ out of $24$ .

$\cot (x) = \frac{2 (12)}{- 10}$

Step 8.3.1.2

Cancel the common factors.

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Step 8.3.1.2.1

Factor $2$ out of $- 10$ .

$\cot (x) = \frac{2 \cdot 12}{2 \cdot - 5}$

Step 8.3.1.2.2

Cancel the common factor.

$\cot (x) = \frac{2 \cdot 12}{2 \cdot - 5}$

Step 8.3.1.2.3

Rewrite the expression.

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

Step 8.3.2

Move the negative in front of the fraction.

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

Step 9

Find the value of secant.

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Step 9.1

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

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

Step 9.2

Substitute in the known values.

$\sec (x) = \frac{26}{24}$

Step 9.3

Cancel the common factor of $26$ and $24$ .

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Step 9.3.1

Factor $2$ out of $26$ .

$\sec (x) = \frac{2 (13)}{24}$

Step 9.3.2

Cancel the common factors.

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Step 9.3.2.1

Factor $2$ out of $24$ .

$\sec (x) = \frac{2 \cdot 13}{2 \cdot 12}$

Step 9.3.2.2

Cancel the common factor.

$\sec (x) = \frac{2 \cdot 13}{2 \cdot 12}$

Step 9.3.2.3

Rewrite the expression.

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

Step 10

Find the value of cosecant.

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Step 10.1

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

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

Step 10.2

Substitute in the known values.

$\csc (x) = \frac{26}{- 10}$

Step 10.3

Simplify the value of $\csc (x)$ .

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Step 10.3.1

Cancel the common factor of $26$ and $- 10$ .

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Step 10.3.1.1

Factor $2$ out of $26$ .

$\csc (x) = \frac{2 (13)}{- 10}$

Step 10.3.1.2

Cancel the common factors.

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Step 10.3.1.2.1

Factor $2$ out of $- 10$ .

$\csc (x) = \frac{2 \cdot 13}{2 \cdot - 5}$

Step 10.3.1.2.2

Cancel the common factor.

$\csc (x) = \frac{2 \cdot 13}{2 \cdot - 5}$

Step 10.3.1.2.3

Rewrite the expression.

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

Step 10.3.2

Move the negative in front of the fraction.

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

Step 11

This is the solution to each trig value.

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

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

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

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

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

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