Precalculus Examples

\tan (θ) = \frac{4}{3}

$\tan (θ) = \frac{4}{3}$

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 (θ) = \frac{opposite}{adjacent}

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

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{{(4)}^{2} + {(3)}^{2}}

$Hypotenuse = \sqrt{{(4)}^{2} + {(3)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Raise

4

$4$ to the power of

2

$2$ .

Hypotenuse

= \sqrt{16 + {(3)}^{2}}

$= \sqrt{16 + {(3)}^{2}}$

Step 4.2

Raise

3

$3$ to the power of

2

$2$ .

Hypotenuse

= \sqrt{16 + 9}

$= \sqrt{16 + 9}$

Step 4.3

Add

16

$16$ and

9

$9$ .

Hypotenuse

= \sqrt{25}

$= \sqrt{25}$

Step 4.4

Rewrite

25

$25$ as

5^{2}

$5^{2}$ .

Hypotenuse

= \sqrt{5^{2}}

$= \sqrt{5^{2}}$

Step 4.5

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

Hypotenuse

= 5

$= 5$

Hypotenuse

= 5

$= 5$

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{4}{5}

$\sin (θ) = \frac{4}{5}$

\sin (θ) = \frac{4}{5}

$\sin (θ) = \frac{4}{5}$

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 (θ)

$\cos (θ)$ .

\cos (θ) = \frac{adj}{hyp}

$\cos (θ) = \frac{adj}{hyp}$

Step 6.2

Substitute in the known values.

\cos (θ) = \frac{3}{5}

$\cos (θ) = \frac{3}{5}$

\cos (θ) = \frac{3}{5}

$\cos (θ) = \frac{3}{5}$

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{3}{4}

$\cot (θ) = \frac{3}{4}$

\cot (θ) = \frac{3}{4}

$\cot (θ) = \frac{3}{4}$

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 (θ)

$\sec (θ)$ .

\sec (θ) = \frac{hyp}{adj}

$\sec (θ) = \frac{hyp}{adj}$

Step 8.2

Substitute in the known values.

\sec (θ) = \frac{5}{3}

$\sec (θ) = \frac{5}{3}$

\sec (θ) = \frac{5}{3}

$\sec (θ) = \frac{5}{3}$

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 (θ)

$\csc (θ)$ .

\csc (θ) = \frac{hyp}{opp}

$\csc (θ) = \frac{hyp}{opp}$

Step 9.2

Substitute in the known values.

\csc (θ) = \frac{5}{4}

$\csc (θ) = \frac{5}{4}$

\csc (θ) = \frac{5}{4}

$\csc (θ) = \frac{5}{4}$

Step 10

This is the solution to each trig value.

\sin (θ) = \frac{4}{5}

$\sin (θ) = \frac{4}{5}$

\cos (θ) = \frac{3}{5}

$\cos (θ) = \frac{3}{5}$

\tan (θ) = \frac{4}{3}

$\tan (θ) = \frac{4}{3}$

\cot (θ) = \frac{3}{4}

$\cot (θ) = \frac{3}{4}$

\sec (θ) = \frac{5}{3}

$\sec (θ) = \frac{5}{3}$

\csc (θ) = \frac{5}{4}

$\csc (θ) = \frac{5}{4}$