Trigonometry Examples

$\sin (x) = - \frac{7}{11}$

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.

$Adjacent = - \sqrt{{(11)}^{2} - {(- 7)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Negate $\sqrt{{(11)}^{2} - {(- 7)}^{2}}$ .

Adjacent $= - \sqrt{{(11)}^{2} - {(- 7)}^{2}}$

Step 4.2

Raise $11$ to the power of $2$ .

Adjacent $= - \sqrt{121 - {(- 7)}^{2}}$

Step 4.3

Raise $- 7$ to the power of $2$ .

Adjacent $= - \sqrt{121 - 1 \cdot 49}$

Step 4.4

Multiply $- 1$ by $49$ .

Adjacent $= - \sqrt{121 - 49}$

Step 4.5

Subtract $49$ from $121$ .

Adjacent $= - \sqrt{72}$

Step 4.6

Rewrite $72$ as $6^{2} \cdot 2$ .

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Step 4.6.1

Factor $36$ out of $72$ .

Adjacent $= - \sqrt{36 (2)}$

Step 4.6.2

Rewrite $36$ as $6^{2}$ .

Adjacent $= - \sqrt{6^{2} \cdot 2}$

Step 4.7

Pull terms out from under the radical.

Adjacent $= - (6 \sqrt{2})$

Step 4.8

Multiply $6$ by $- 1$ .

Adjacent $= - 6 \sqrt{2}$

Step 5

Find the value of cosine.

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Step 5.1

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

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

Step 5.2

Substitute in the known values.

$\cos (x) = \frac{- 6 \sqrt{2}}{11}$

Step 5.3

Move the negative in front of the fraction.

$\cos (x) = - \frac{6 \sqrt{2}}{11}$

Step 6

Find the value of tangent.

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Step 6.1

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

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

Step 6.2

Substitute in the known values.

$\tan (x) = \frac{- 7}{- 6 \sqrt{2}}$

Step 6.3

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

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Step 6.3.1

Dividing two negative values results in a positive value.

$\tan (x) = \frac{7}{6 \sqrt{2}}$

Step 6.3.2

Multiply $\frac{7}{6 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\tan (x) = \frac{7}{6 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 6.3.3

Combine and simplify the denominator.

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Step 6.3.3.1

Multiply $\frac{7}{6 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\tan (x) = \frac{7 \sqrt{2}}{6 \sqrt{2} \sqrt{2}}$

Step 6.3.3.2

Move $\sqrt{2}$ .

$\tan (x) = \frac{7 \sqrt{2}}{6 (\sqrt{2} \sqrt{2})}$

Step 6.3.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$\tan (x) = \frac{7 \sqrt{2}}{6 (\sqrt{2} \sqrt{2})}$

Step 6.3.3.4

Raise $\sqrt{2}$ to the power of $1$ .

$\tan (x) = \frac{7 \sqrt{2}}{6 (\sqrt{2} \sqrt{2})}$

Step 6.3.3.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\tan (x) = \frac{7 \sqrt{2}}{6 {\sqrt{2}}^{1 + 1}}$

Step 6.3.3.6

Add $1$ and $1$ .

$\tan (x) = \frac{7 \sqrt{2}}{6 {\sqrt{2}}^{2}}$

Step 6.3.3.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 6.3.3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\tan (x) = \frac{7 \sqrt{2}}{6 {(2^{\frac{1}{2}})}^{2}}$

Step 6.3.3.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\tan (x) = \frac{7 \sqrt{2}}{6 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 6.3.3.7.3

Combine $\frac{1}{2}$ and $2$ .

$\tan (x) = \frac{7 \sqrt{2}}{6 \cdot 2^{\frac{2}{2}}}$

Step 6.3.3.7.4

Cancel the common factor of $2$ .

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Step 6.3.3.7.4.1

Cancel the common factor.

$\tan (x) = \frac{7 \sqrt{2}}{6 \cdot 2^{\frac{2}{2}}}$

Step 6.3.3.7.4.2

Rewrite the expression.

$\tan (x) = \frac{7 \sqrt{2}}{6 \cdot 2}$

Step 6.3.3.7.5

Evaluate the exponent.

$\tan (x) = \frac{7 \sqrt{2}}{6 \cdot 2}$

Step 6.3.4

Multiply $6$ by $2$ .

$\tan (x) = \frac{7 \sqrt{2}}{12}$

Step 7

Find the value of cotangent.

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Step 7.1

Use the definition of cotangent to find the value of $\cot (x)$ .

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

Step 7.2

Substitute in the known values.

$\cot (x) = \frac{- 6 \sqrt{2}}{- 7}$

Step 7.3

Dividing two negative values results in a positive value.

$\cot (x) = \frac{6 \sqrt{2}}{7}$

Step 8

Find the value of secant.

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Step 8.1

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

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

Step 8.2

Substitute in the known values.

$\sec (x) = \frac{11}{- 6 \sqrt{2}}$

Step 8.3

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

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Step 8.3.1

Move the negative in front of the fraction.

$\sec (x) = - \frac{11}{6 \sqrt{2}}$

Step 8.3.2

Multiply $\frac{11}{6 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sec (x) = - (\frac{11}{6 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 8.3.3

Combine and simplify the denominator.

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Step 8.3.3.1

Multiply $\frac{11}{6 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sec (x) = - \frac{11 \sqrt{2}}{6 \sqrt{2} \sqrt{2}}$

Step 8.3.3.2

Move $\sqrt{2}$ .

$\sec (x) = - \frac{11 \sqrt{2}}{6 (\sqrt{2} \sqrt{2})}$

Step 8.3.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$\sec (x) = - \frac{11 \sqrt{2}}{6 (\sqrt{2} \sqrt{2})}$

Step 8.3.3.4

Raise $\sqrt{2}$ to the power of $1$ .

$\sec (x) = - \frac{11 \sqrt{2}}{6 (\sqrt{2} \sqrt{2})}$

Step 8.3.3.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sec (x) = - \frac{11 \sqrt{2}}{6 {\sqrt{2}}^{1 + 1}}$

Step 8.3.3.6

Add $1$ and $1$ .

$\sec (x) = - \frac{11 \sqrt{2}}{6 {\sqrt{2}}^{2}}$

Step 8.3.3.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 8.3.3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\sec (x) = - \frac{11 \sqrt{2}}{6 {(2^{\frac{1}{2}})}^{2}}$

Step 8.3.3.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sec (x) = - \frac{11 \sqrt{2}}{6 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 8.3.3.7.3

Combine $\frac{1}{2}$ and $2$ .

$\sec (x) = - \frac{11 \sqrt{2}}{6 \cdot 2^{\frac{2}{2}}}$

Step 8.3.3.7.4

Cancel the common factor of $2$ .

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Step 8.3.3.7.4.1

Cancel the common factor.

$\sec (x) = - \frac{11 \sqrt{2}}{6 \cdot 2^{\frac{2}{2}}}$

Step 8.3.3.7.4.2

Rewrite the expression.

$\sec (x) = - \frac{11 \sqrt{2}}{6 \cdot 2}$

Step 8.3.3.7.5

Evaluate the exponent.

$\sec (x) = - \frac{11 \sqrt{2}}{6 \cdot 2}$

Step 8.3.4

Multiply $6$ by $2$ .

$\sec (x) = - \frac{11 \sqrt{2}}{12}$

Step 9

Find the value of cosecant.

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Step 9.1

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

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

Step 9.2

Substitute in the known values.

$\csc (x) = \frac{11}{- 7}$

Step 9.3

Move the negative in front of the fraction.

$\csc (x) = - \frac{11}{7}$

Step 10

This is the solution to each trig value.

$\sin (x) = - \frac{7}{11}$

$\cos (x) = - \frac{6 \sqrt{2}}{11}$

$\tan (x) = \frac{7 \sqrt{2}}{12}$

$\cot (x) = \frac{6 \sqrt{2}}{7}$

$\sec (x) = - \frac{11 \sqrt{2}}{12}$

$\csc (x) = - \frac{11}{7}$