Trigonometry Examples

$\sin (x) = \frac{12}{15}$

Step 1

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

$\sin (x) = \frac{opposite}{hypotenuse}$

Step 2

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

$Adjacent = \sqrt{{hypotenuse}^{2} - {opposite}^{2}}$

Step 3

Replace the known values in the equation.

$Adjacent = \sqrt{{(15)}^{2} - {(12)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Raise $15$ to the power of $2$ .

Adjacent $= \sqrt{225 - {(12)}^{2}}$

Step 4.2

Raise $12$ to the power of $2$ .

Adjacent $= \sqrt{225 - 1 \cdot 144}$

Step 4.3

Multiply $- 1$ by $144$ .

Adjacent $= \sqrt{225 - 144}$

Step 4.4

Subtract $144$ from $225$ .

Adjacent $= \sqrt{81}$

Step 4.5

Rewrite $81$ as $9^{2}$ .

Adjacent $= \sqrt{9^{2}}$

Step 4.6

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

Adjacent $= 9$

Step 5

Cancel the common factor of $12$ and $15$ .

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Step 5.1

Factor $3$ out of $12$ .

$\sin (x) = \frac{3 (4)}{15}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $3$ out of $15$ .

$\sin (x) = \frac{3 \cdot 4}{3 \cdot 5}$

Step 5.2.2

Cancel the common factor.

$\sin (x) = \frac{3 \cdot 4}{3 \cdot 5}$

Step 5.2.3

Rewrite the expression.

$\sin (x) = \frac{4}{5}$

Step 6

Find the value of cosine.

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Step 6.1

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

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

Step 6.2

Substitute in the known values.

$\cos (x) = \frac{9}{15}$

Step 6.3

Cancel the common factor of $9$ and $15$ .

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Step 6.3.1

Factor $3$ out of $9$ .

$\cos (x) = \frac{3 (3)}{15}$

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Factor $3$ out of $15$ .

$\cos (x) = \frac{3 \cdot 3}{3 \cdot 5}$

Step 6.3.2.2

Cancel the common factor.

$\cos (x) = \frac{3 \cdot 3}{3 \cdot 5}$

Step 6.3.2.3

Rewrite the expression.

$\cos (x) = \frac{3}{5}$

Step 7

Find the value of tangent.

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Step 7.1

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

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

Step 7.2

Substitute in the known values.

$\tan (x) = \frac{12}{9}$

Step 7.3

Cancel the common factor of $12$ and $9$ .

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Step 7.3.1

Factor $3$ out of $12$ .

$\tan (x) = \frac{3 (4)}{9}$

Step 7.3.2

Cancel the common factors.

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Step 7.3.2.1

Factor $3$ out of $9$ .

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

Step 7.3.2.2

Cancel the common factor.

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

Step 7.3.2.3

Rewrite the expression.

$\tan (x) = \frac{4}{3}$

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.

$\cot (x) = \frac{9}{12}$

Step 8.3

Cancel the common factor of $9$ and $12$ .

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Step 8.3.1

Factor $3$ out of $9$ .

$\cot (x) = \frac{3 (3)}{12}$

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

Factor $3$ out of $12$ .

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

Step 8.3.2.2

Cancel the common factor.

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

Step 8.3.2.3

Rewrite the expression.

$\cot (x) = \frac{3}{4}$

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.

$\sec (x) = \frac{15}{9}$

Step 9.3

Cancel the common factor of $15$ and $9$ .

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Step 9.3.1

Factor $3$ out of $15$ .

$\sec (x) = \frac{3 (5)}{9}$

Step 9.3.2

Cancel the common factors.

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Step 9.3.2.1

Factor $3$ out of $9$ .

$\sec (x) = \frac{3 \cdot 5}{3 \cdot 3}$

Step 9.3.2.2

Cancel the common factor.

$\sec (x) = \frac{3 \cdot 5}{3 \cdot 3}$

Step 9.3.2.3

Rewrite the expression.

$\sec (x) = \frac{5}{3}$

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.

$\csc (x) = \frac{15}{12}$

Step 10.3

Cancel the common factor of $15$ and $12$ .

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Step 10.3.1

Factor $3$ out of $15$ .

$\csc (x) = \frac{3 (5)}{12}$

Step 10.3.2

Cancel the common factors.

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Step 10.3.2.1

Factor $3$ out of $12$ .

$\csc (x) = \frac{3 \cdot 5}{3 \cdot 4}$

Step 10.3.2.2

Cancel the common factor.

$\csc (x) = \frac{3 \cdot 5}{3 \cdot 4}$

Step 10.3.2.3

Rewrite the expression.

$\csc (x) = \frac{5}{4}$

Step 11

This is the solution to each trig value.

$\sin (x) = \frac{4}{5}$

$\cos (x) = \frac{3}{5}$

$\tan (x) = \frac{4}{3}$

$\cot (x) = \frac{3}{4}$

$\sec (x) = \frac{5}{3}$

$\csc (x) = \frac{5}{4}$