Precalculus Examples

$\cot (θ) = \frac{8}{6}$

Step 1

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

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

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.

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

Step 3

Replace the known values in the equation.

$Hypotenuse = \sqrt{{(6)}^{2} + {(8)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Raise $6$ to the power of $2$ .

Hypotenuse $= \sqrt{36 + {(8)}^{2}}$

Step 4.2

Raise $8$ to the power of $2$ .

Hypotenuse $= \sqrt{36 + 64}$

Step 4.3

Add $36$ and $64$ .

Hypotenuse $= \sqrt{100}$

Step 4.4

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

Hypotenuse $= \sqrt{10^{2}}$

Step 4.5

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

Hypotenuse $= 10$

Step 5

Find the value of sine.

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Step 5.1

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

$\sin (θ) = \frac{opp}{hyp}$

Step 5.2

Substitute in the known values.

$\sin (θ) = \frac{6}{10}$

Step 5.3

Cancel the common factor of $6$ and $10$ .

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Step 5.3.1

Factor $2$ out of $6$ .

$\sin (θ) = \frac{2 (3)}{10}$

Step 5.3.2

Cancel the common factors.

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Step 5.3.2.1

Factor $2$ out of $10$ .

$\sin (θ) = \frac{2 \cdot 3}{2 \cdot 5}$

Step 5.3.2.2

Cancel the common factor.

$\sin (θ) = \frac{2 \cdot 3}{2 \cdot 5}$

Step 5.3.2.3

Rewrite the expression.

$\sin (θ) = \frac{3}{5}$

Step 6

Find the value of cosine.

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Step 6.1

Use the definition of cosine to find the value of $\cos (θ)$ .

$\cos (θ) = \frac{adj}{hyp}$

Step 6.2

Substitute in the known values.

$\cos (θ) = \frac{8}{10}$

Step 6.3

Cancel the common factor of $8$ and $10$ .

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Step 6.3.1

Factor $2$ out of $8$ .

$\cos (θ) = \frac{2 (4)}{10}$

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Factor $2$ out of $10$ .

$\cos (θ) = \frac{2 \cdot 4}{2 \cdot 5}$

Step 6.3.2.2

Cancel the common factor.

$\cos (θ) = \frac{2 \cdot 4}{2 \cdot 5}$

Step 6.3.2.3

Rewrite the expression.

$\cos (θ) = \frac{4}{5}$

Step 7

Find the value of tangent.

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Step 7.1

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

$\tan (θ) = \frac{opp}{adj}$

Step 7.2

Substitute in the known values.

$\tan (θ) = \frac{6}{8}$

Step 7.3

Cancel the common factor of $6$ and $8$ .

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Step 7.3.1

Factor $2$ out of $6$ .

$\tan (θ) = \frac{2 (3)}{8}$

Step 7.3.2

Cancel the common factors.

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Step 7.3.2.1

Factor $2$ out of $8$ .

$\tan (θ) = \frac{2 \cdot 3}{2 \cdot 4}$

Step 7.3.2.2

Cancel the common factor.

$\tan (θ) = \frac{2 \cdot 3}{2 \cdot 4}$

Step 7.3.2.3

Rewrite the expression.

$\tan (θ) = \frac{3}{4}$

Step 8

Cancel the common factor of $8$ and $6$ .

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Step 8.1

Factor $2$ out of $8$ .

$\cot (θ) = \frac{2 (4)}{6}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $2$ out of $6$ .

$\cot (θ) = \frac{2 \cdot 4}{2 \cdot 3}$

Step 8.2.2

Cancel the common factor.

$\cot (θ) = \frac{2 \cdot 4}{2 \cdot 3}$

Step 8.2.3

Rewrite the expression.

$\cot (θ) = \frac{4}{3}$

Step 9

Find the value of secant.

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Step 9.1

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

$\sec (θ) = \frac{hyp}{adj}$

Step 9.2

Substitute in the known values.

$\sec (θ) = \frac{10}{8}$

Step 9.3

Cancel the common factor of $10$ and $8$ .

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Step 9.3.1

Factor $2$ out of $10$ .

$\sec (θ) = \frac{2 (5)}{8}$

Step 9.3.2

Cancel the common factors.

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Step 9.3.2.1

Factor $2$ out of $8$ .

$\sec (θ) = \frac{2 \cdot 5}{2 \cdot 4}$

Step 9.3.2.2

Cancel the common factor.

$\sec (θ) = \frac{2 \cdot 5}{2 \cdot 4}$

Step 9.3.2.3

Rewrite the expression.

$\sec (θ) = \frac{5}{4}$

Step 10

Find the value of cosecant.

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Step 10.1

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

$\csc (θ) = \frac{hyp}{opp}$

Step 10.2

Substitute in the known values.

$\csc (θ) = \frac{10}{6}$

Step 10.3

Cancel the common factor of $10$ and $6$ .

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Step 10.3.1

Factor $2$ out of $10$ .

$\csc (θ) = \frac{2 (5)}{6}$

Step 10.3.2

Cancel the common factors.

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Step 10.3.2.1

Factor $2$ out of $6$ .

$\csc (θ) = \frac{2 \cdot 5}{2 \cdot 3}$

Step 10.3.2.2

Cancel the common factor.

$\csc (θ) = \frac{2 \cdot 5}{2 \cdot 3}$

Step 10.3.2.3

Rewrite the expression.

$\csc (θ) = \frac{5}{3}$

Step 11

This is the solution to each trig value.

$\sin (θ) = \frac{3}{5}$

$\cos (θ) = \frac{4}{5}$

$\tan (θ) = \frac{3}{4}$

$\cot (θ) = \frac{4}{3}$

$\sec (θ) = \frac{5}{4}$

$\csc (θ) = \frac{5}{3}$