Trigonometry Examples

$\cos (x) = \frac{6}{10}$

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.

$Opposite = - \sqrt{{(10)}^{2} - {(6)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Negate $\sqrt{{(10)}^{2} - {(6)}^{2}}$ .

Opposite $= - \sqrt{{(10)}^{2} - {(6)}^{2}}$

Step 4.2

Raise $10$ to the power of $2$ .

Opposite $= - \sqrt{100 - {(6)}^{2}}$

Step 4.3

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

Opposite $= - \sqrt{100 - 1 \cdot 36}$

Step 4.4

Multiply $- 1$ by $36$ .

Opposite $= - \sqrt{100 - 36}$

Step 4.5

Subtract $36$ from $100$ .

Opposite $= - \sqrt{64}$

Step 4.6

Rewrite $64$ as $8^{2}$ .

Opposite $= - \sqrt{8^{2}}$

Step 4.7

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

Opposite $= - 1 \cdot 8$

Step 4.8

Multiply $- 1$ by $8$ .

Opposite $= - 8$

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.

$\sin (x) = \frac{- 8}{10}$

Step 5.3

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

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Step 5.3.1

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

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Step 5.3.1.1

Factor $2$ out of $- 8$ .

$\sin (x) = \frac{2 (- 4)}{10}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $2$ out of $10$ .

$\sin (x) = \frac{2 \cdot - 4}{2 \cdot 5}$

Step 5.3.1.2.2

Cancel the common factor.

$\sin (x) = \frac{2 \cdot - 4}{2 \cdot 5}$

Step 5.3.1.2.3

Rewrite the expression.

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

Step 5.3.2

Move the negative in front of the fraction.

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

Step 6

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

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Step 6.1

Factor $2$ out of $6$ .

$\cos (x) = \frac{2 (3)}{10}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $2$ out of $10$ .

$\cos (x) = \frac{2 \cdot 3}{2 \cdot 5}$

Step 6.2.2

Cancel the common factor.

$\cos (x) = \frac{2 \cdot 3}{2 \cdot 5}$

Step 6.2.3

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.

$\tan (x) = \frac{- 8}{6}$

Step 7.3

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

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Step 7.3.1

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

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Step 7.3.1.1

Factor $2$ out of $- 8$ .

$\tan (x) = \frac{2 (- 4)}{6}$

Step 7.3.1.2

Cancel the common factors.

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Step 7.3.1.2.1

Factor $2$ out of $6$ .

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

Step 7.3.1.2.2

Cancel the common factor.

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

Step 7.3.1.2.3

Rewrite the expression.

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

Step 7.3.2

Move the negative in front of the fraction.

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

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.

$\cot (x) = \frac{6}{- 8}$

Step 8.3

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

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Step 8.3.1

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

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Step 8.3.1.1

Factor $2$ out of $6$ .

$\cot (x) = \frac{2 (3)}{- 8}$

Step 8.3.1.2

Cancel the common factors.

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Step 8.3.1.2.1

Factor $2$ out of $- 8$ .

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

Step 8.3.1.2.2

Cancel the common factor.

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

Step 8.3.1.2.3

Rewrite the expression.

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

Step 8.3.2

Move the negative in front of the fraction.

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

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.

$\sec (x) = \frac{10}{6}$

Step 9.3

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

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Step 9.3.1

Factor $2$ out of $10$ .

$\sec (x) = \frac{2 (5)}{6}$

Step 9.3.2

Cancel the common factors.

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Step 9.3.2.1

Factor $2$ out of $6$ .

$\sec (x) = \frac{2 \cdot 5}{2 \cdot 3}$

Step 9.3.2.2

Cancel the common factor.

$\sec (x) = \frac{2 \cdot 5}{2 \cdot 3}$

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.

$\csc (x) = \frac{10}{- 8}$

Step 10.3

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

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Step 10.3.1

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

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Step 10.3.1.1

Factor $2$ out of $10$ .

$\csc (x) = \frac{2 (5)}{- 8}$

Step 10.3.1.2

Cancel the common factors.

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Step 10.3.1.2.1

Factor $2$ out of $- 8$ .

$\csc (x) = \frac{2 \cdot 5}{2 \cdot - 4}$

Step 10.3.1.2.2

Cancel the common factor.

$\csc (x) = \frac{2 \cdot 5}{2 \cdot - 4}$

Step 10.3.1.2.3

Rewrite the expression.

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

Step 10.3.2

Move the negative in front of the fraction.

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

Step 11

This is the solution to each trig value.

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

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

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

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

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

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