Trigonometry Examples

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

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.

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

Step 4

Simplify inside the radical.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

Adjacent $= \sqrt{36 - {(2^{\frac{1}{2}})}^{2}}$

Step 4.2.2

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

Adjacent $= \sqrt{36 - 2^{\frac{1}{2} \cdot 2}}$

Step 4.2.3

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

Adjacent $= \sqrt{36 - 2^{\frac{2}{2}}}$

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Cancel the common factor.

Adjacent $= \sqrt{36 - 2^{\frac{2}{2}}}$

Step 4.2.4.2

Rewrite the expression.

Adjacent $= \sqrt{36 - 2}$

Step 4.2.5

Evaluate the exponent.

Adjacent $= \sqrt{36 - 1 \cdot 2}$

Step 4.3

Multiply $- 1$ by $2$ .

Adjacent $= \sqrt{36 - 2}$

Step 4.4

Subtract $2$ from $36$ .

Adjacent $= \sqrt{34}$

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.

$\cos (x) = \frac{\sqrt{34}}{6}$

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.

$\tan (x) = \frac{\sqrt{2}}{\sqrt{34}}$

Step 6.3

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

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Step 6.3.1

Combine $\sqrt{2}$ and $\sqrt{34}$ into a single radical.

$\tan (x) = \sqrt{\frac{2}{34}}$

Step 6.3.2

Cancel the common factor of $2$ and $34$ .

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Step 6.3.2.1

Factor $2$ out of $2$ .

$\tan (x) = \sqrt{\frac{2 (1)}{34}}$

Step 6.3.2.2

Cancel the common factors.

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Step 6.3.2.2.1

Factor $2$ out of $34$ .

$\tan (x) = \sqrt{\frac{2 \cdot 1}{2 \cdot 17}}$

Step 6.3.2.2.2

Cancel the common factor.

$\tan (x) = \sqrt{\frac{2 \cdot 1}{2 \cdot 17}}$

Step 6.3.2.2.3

Rewrite the expression.

$\tan (x) = \sqrt{\frac{1}{17}}$

Step 6.3.3

Rewrite $\sqrt{\frac{1}{17}}$ as $\frac{\sqrt{1}}{\sqrt{17}}$ .

$\tan (x) = \frac{\sqrt{1}}{\sqrt{17}}$

Step 6.3.4

Any root of $1$ is $1$ .

$\tan (x) = \frac{1}{\sqrt{17}}$

Step 6.3.5

Multiply $\frac{1}{\sqrt{17}}$ by $\frac{\sqrt{17}}{\sqrt{17}}$ .

$\tan (x) = \frac{1}{\sqrt{17}} \cdot \frac{\sqrt{17}}{\sqrt{17}}$

Step 6.3.6

Combine and simplify the denominator.

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Step 6.3.6.1

Multiply $\frac{1}{\sqrt{17}}$ by $\frac{\sqrt{17}}{\sqrt{17}}$ .

$\tan (x) = \frac{\sqrt{17}}{\sqrt{17} \sqrt{17}}$

Step 6.3.6.2

Raise $\sqrt{17}$ to the power of $1$ .

$\tan (x) = \frac{\sqrt{17}}{\sqrt{17} \sqrt{17}}$

Step 6.3.6.3

Raise $\sqrt{17}$ to the power of $1$ .

$\tan (x) = \frac{\sqrt{17}}{\sqrt{17} \sqrt{17}}$

Step 6.3.6.4

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

$\tan (x) = \frac{\sqrt{17}}{{\sqrt{17}}^{1 + 1}}$

Step 6.3.6.5

Add $1$ and $1$ .

$\tan (x) = \frac{\sqrt{17}}{{\sqrt{17}}^{2}}$

Step 6.3.6.6

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

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Step 6.3.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$\tan (x) = \frac{\sqrt{17}}{{(17^{\frac{1}{2}})}^{2}}$

Step 6.3.6.6.2

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

$\tan (x) = \frac{\sqrt{17}}{17^{\frac{1}{2} \cdot 2}}$

Step 6.3.6.6.3

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

$\tan (x) = \frac{\sqrt{17}}{17^{\frac{2}{2}}}$

Step 6.3.6.6.4

Cancel the common factor of $2$ .

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Step 6.3.6.6.4.1

Cancel the common factor.

$\tan (x) = \frac{\sqrt{17}}{17^{\frac{2}{2}}}$

Step 6.3.6.6.4.2

Rewrite the expression.

$\tan (x) = \frac{\sqrt{17}}{17}$

Step 6.3.6.6.5

Evaluate the exponent.

$\tan (x) = \frac{\sqrt{17}}{17}$

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.

$\cot (x) = \frac{\sqrt{34}}{\sqrt{2}}$

Step 7.3

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

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Step 7.3.1

Combine $\sqrt{34}$ and $\sqrt{2}$ into a single radical.

$\cot (x) = \sqrt{\frac{34}{2}}$

Step 7.3.2

Divide $34$ by $2$ .

$\cot (x) = \sqrt{17}$

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.

$\sec (x) = \frac{6}{\sqrt{34}}$

Step 8.3

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

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Step 8.3.1

Multiply $\frac{6}{\sqrt{34}}$ by $\frac{\sqrt{34}}{\sqrt{34}}$ .

$\sec (x) = \frac{6}{\sqrt{34}} \cdot \frac{\sqrt{34}}{\sqrt{34}}$

Step 8.3.2

Combine and simplify the denominator.

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Step 8.3.2.1

Multiply $\frac{6}{\sqrt{34}}$ by $\frac{\sqrt{34}}{\sqrt{34}}$ .

$\sec (x) = \frac{6 \sqrt{34}}{\sqrt{34} \sqrt{34}}$

Step 8.3.2.2

Raise $\sqrt{34}$ to the power of $1$ .

$\sec (x) = \frac{6 \sqrt{34}}{\sqrt{34} \sqrt{34}}$

Step 8.3.2.3

Raise $\sqrt{34}$ to the power of $1$ .

$\sec (x) = \frac{6 \sqrt{34}}{\sqrt{34} \sqrt{34}}$

Step 8.3.2.4

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

$\sec (x) = \frac{6 \sqrt{34}}{{\sqrt{34}}^{1 + 1}}$

Step 8.3.2.5

Add $1$ and $1$ .

$\sec (x) = \frac{6 \sqrt{34}}{{\sqrt{34}}^{2}}$

Step 8.3.2.6

Rewrite ${\sqrt{34}}^{2}$ as $34$ .

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Step 8.3.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$\sec (x) = \frac{6 \sqrt{34}}{{(34^{\frac{1}{2}})}^{2}}$

Step 8.3.2.6.2

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

$\sec (x) = \frac{6 \sqrt{34}}{34^{\frac{1}{2} \cdot 2}}$

Step 8.3.2.6.3

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

$\sec (x) = \frac{6 \sqrt{34}}{34^{\frac{2}{2}}}$

Step 8.3.2.6.4

Cancel the common factor of $2$ .

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Step 8.3.2.6.4.1

Cancel the common factor.

$\sec (x) = \frac{6 \sqrt{34}}{34^{\frac{2}{2}}}$

Step 8.3.2.6.4.2

Rewrite the expression.

$\sec (x) = \frac{6 \sqrt{34}}{34}$

Step 8.3.2.6.5

Evaluate the exponent.

$\sec (x) = \frac{6 \sqrt{34}}{34}$

Step 8.3.3

Cancel the common factor of $6$ and $34$ .

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Step 8.3.3.1

Factor $2$ out of $6 \sqrt{34}$ .

$\sec (x) = \frac{2 (3 \sqrt{34})}{34}$

Step 8.3.3.2

Cancel the common factors.

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Step 8.3.3.2.1

Factor $2$ out of $34$ .

$\sec (x) = \frac{2 (3 \sqrt{34})}{2 (17)}$

Step 8.3.3.2.2

Cancel the common factor.

$\sec (x) = \frac{2 (3 \sqrt{34})}{2 \cdot 17}$

Step 8.3.3.2.3

Rewrite the expression.

$\sec (x) = \frac{3 \sqrt{34}}{17}$

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.

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

Step 9.3

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

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Step 9.3.1

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

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

Step 9.3.2

Combine and simplify the denominator.

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Step 9.3.2.1

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

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

Step 9.3.2.2

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

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

Step 9.3.2.3

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

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

Step 9.3.2.4

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

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

Step 9.3.2.5

Add $1$ and $1$ .

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

Step 9.3.2.6

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

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Step 9.3.2.6.1

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

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

Step 9.3.2.6.2

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

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

Step 9.3.2.6.3

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

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

Step 9.3.2.6.4

Cancel the common factor of $2$ .

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Step 9.3.2.6.4.1

Cancel the common factor.

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

Step 9.3.2.6.4.2

Rewrite the expression.

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

Step 9.3.2.6.5

Evaluate the exponent.

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

Step 9.3.3

Cancel the common factor of $6$ and $2$ .

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Step 9.3.3.1

Factor $2$ out of $6 \sqrt{2}$ .

$\csc (x) = \frac{2 (3 \sqrt{2})}{2}$

Step 9.3.3.2

Cancel the common factors.

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Step 9.3.3.2.1

Factor $2$ out of $2$ .

$\csc (x) = \frac{2 (3 \sqrt{2})}{2 (1)}$

Step 9.3.3.2.2

Cancel the common factor.

$\csc (x) = \frac{2 (3 \sqrt{2})}{2 \cdot 1}$

Step 9.3.3.2.3

Rewrite the expression.

$\csc (x) = \frac{3 \sqrt{2}}{1}$

Step 9.3.3.2.4

Divide $3 \sqrt{2}$ by $1$ .

$\csc (x) = 3 \sqrt{2}$

Step 10

This is the solution to each trig value.

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

$\cos (x) = \frac{\sqrt{34}}{6}$

$\tan (x) = \frac{\sqrt{17}}{17}$

$\cot (x) = \sqrt{17}$

$\sec (x) = \frac{3 \sqrt{34}}{17}$

$\csc (x) = 3 \sqrt{2}$