Algebra Examples

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

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

Step 1

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

tan (x) = \frac{opposite}{adjacent}

$\tan (x) = \frac{opposite}{adjacent}$

Step 2

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

Hypotenuse = \sqrt{{opposite}^{2} + {adjacent}^{2}}

$Hypotenuse = \sqrt{{opposite}^{2} + {adjacent}^{2}}$

Step 3

Replace the known values in the equation.

Hypotenuse = \sqrt{{(12)}^{2} + {(5)}^{2}}

$Hypotenuse = \sqrt{{(12)}^{2} + {(5)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Raise

12

$12$ to the power of

2

$2$ .

Hypotenuse

= \sqrt{144 + {(5)}^{2}}

$= \sqrt{144 + {(5)}^{2}}$

Step 4.2

Raise

5

$5$ to the power of

2

$2$ .

Hypotenuse

= \sqrt{144 + 25}

$= \sqrt{144 + 25}$

Step 4.3

Add

144

$144$ and

25

$25$ .

Hypotenuse

= \sqrt{169}

$= \sqrt{169}$

Step 4.4

Rewrite

169

$169$ as

13^{2}

$13^{2}$ .

Hypotenuse

= \sqrt{13^{2}}

$= \sqrt{13^{2}}$

Step 4.5

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

Hypotenuse

= 13

$= 13$

Hypotenuse

= 13

$= 13$

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)

$\sin (x)$ .

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

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

Step 5.2

Substitute in the known values.

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

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

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

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

Step 6

Find the value of cosine.

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Step 6.1

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 6.2

Substitute in the known values.

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

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

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

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

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{5}{12}

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

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

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

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)

$\sec (x)$ .

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

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

Step 8.2

Substitute in the known values.

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

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

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

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

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)

$\csc (x)$ .

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

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

Step 9.2

Substitute in the known values.

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

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

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

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

Step 10

This is the solution to each trig value.

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

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

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

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

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

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

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

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

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

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

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

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