Trigonometry Examples

$\sin (x) = \frac{\sqrt{7}}{7}$

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.

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

Step 4

Simplify inside the radical.

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Step 4.1

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

Adjacent $= \sqrt{49 - {(\sqrt{7})}^{2}}$

Step 4.2

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

Adjacent $= \sqrt{49 - {(7^{\frac{1}{2}})}^{2}}$

Step 4.2.2

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

Adjacent $= \sqrt{49 - 7^{\frac{1}{2} \cdot 2}}$

Step 4.2.3

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

Adjacent $= \sqrt{49 - 7^{\frac{2}{2}}}$

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Cancel the common factor.

Adjacent $= \sqrt{49 - 7^{\frac{2}{2}}}$

Step 4.2.4.2

Rewrite the expression.

Adjacent $= \sqrt{49 - 7}$

Step 4.2.5

Evaluate the exponent.

Adjacent $= \sqrt{49 - 1 \cdot 7}$

Step 4.3

Multiply $- 1$ by $7$ .

Adjacent $= \sqrt{49 - 7}$

Step 4.4

Subtract $7$ from $49$ .

Adjacent $= \sqrt{42}$

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.

$\cos (x) = \frac{\sqrt{42}}{7}$

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.

$\tan (x) = \frac{\sqrt{7}}{\sqrt{42}}$

Step 6.3

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

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Step 6.3.1

Combine $\sqrt{7}$ and $\sqrt{42}$ into a single radical.

$\tan (x) = \sqrt{\frac{7}{42}}$

Step 6.3.2

Cancel the common factor of $7$ and $42$ .

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Step 6.3.2.1

Factor $7$ out of $7$ .

$\tan (x) = \sqrt{\frac{7 (1)}{42}}$

Step 6.3.2.2

Cancel the common factors.

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Step 6.3.2.2.1

Factor $7$ out of $42$ .

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

Step 6.3.2.2.2

Cancel the common factor.

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

Step 6.3.2.2.3

Rewrite the expression.

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

Step 6.3.3

Rewrite $\sqrt{\frac{1}{6}}$ as $\frac{\sqrt{1}}{\sqrt{6}}$ .

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

Step 6.3.4

Any root of $1$ is $1$ .

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

Step 6.3.5

Multiply $\frac{1}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

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

Step 6.3.6

Combine and simplify the denominator.

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Step 6.3.6.1

Multiply $\frac{1}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

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

Step 6.3.6.2

Raise $\sqrt{6}$ to the power of $1$ .

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

Step 6.3.6.3

Raise $\sqrt{6}$ to the power of $1$ .

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

Step 6.3.6.4

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

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

Step 6.3.6.5

Add $1$ and $1$ .

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

Step 6.3.6.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 6.3.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

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

Step 6.3.6.6.2

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

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

Step 6.3.6.6.3

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

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

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.

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

Step 6.3.6.6.4.2

Rewrite the expression.

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

Step 6.3.6.6.5

Evaluate the exponent.

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

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.

$\cot (x) = \frac{\sqrt{42}}{\sqrt{7}}$

Step 7.3

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

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Step 7.3.1

Combine $\sqrt{42}$ and $\sqrt{7}$ into a single radical.

$\cot (x) = \sqrt{\frac{42}{7}}$

Step 7.3.2

Divide $42$ by $7$ .

$\cot (x) = \sqrt{6}$

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.

$\sec (x) = \frac{7}{\sqrt{42}}$

Step 8.3

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

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Step 8.3.1

Multiply $\frac{7}{\sqrt{42}}$ by $\frac{\sqrt{42}}{\sqrt{42}}$ .

$\sec (x) = \frac{7}{\sqrt{42}} \cdot \frac{\sqrt{42}}{\sqrt{42}}$

Step 8.3.2

Combine and simplify the denominator.

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Step 8.3.2.1

Multiply $\frac{7}{\sqrt{42}}$ by $\frac{\sqrt{42}}{\sqrt{42}}$ .

$\sec (x) = \frac{7 \sqrt{42}}{\sqrt{42} \sqrt{42}}$

Step 8.3.2.2

Raise $\sqrt{42}$ to the power of $1$ .

$\sec (x) = \frac{7 \sqrt{42}}{\sqrt{42} \sqrt{42}}$

Step 8.3.2.3

Raise $\sqrt{42}$ to the power of $1$ .

$\sec (x) = \frac{7 \sqrt{42}}{\sqrt{42} \sqrt{42}}$

Step 8.3.2.4

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

$\sec (x) = \frac{7 \sqrt{42}}{{\sqrt{42}}^{1 + 1}}$

Step 8.3.2.5

Add $1$ and $1$ .

$\sec (x) = \frac{7 \sqrt{42}}{{\sqrt{42}}^{2}}$

Step 8.3.2.6

Rewrite ${\sqrt{42}}^{2}$ as $42$ .

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Step 8.3.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{42}$ as $42^{\frac{1}{2}}$ .

$\sec (x) = \frac{7 \sqrt{42}}{{(42^{\frac{1}{2}})}^{2}}$

Step 8.3.2.6.2

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

$\sec (x) = \frac{7 \sqrt{42}}{42^{\frac{1}{2} \cdot 2}}$

Step 8.3.2.6.3

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

$\sec (x) = \frac{7 \sqrt{42}}{42^{\frac{2}{2}}}$

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.

$\sec (x) = \frac{7 \sqrt{42}}{42^{\frac{2}{2}}}$

Step 8.3.2.6.4.2

Rewrite the expression.

$\sec (x) = \frac{7 \sqrt{42}}{42}$

Step 8.3.2.6.5

Evaluate the exponent.

$\sec (x) = \frac{7 \sqrt{42}}{42}$

Step 8.3.3

Cancel the common factor of $7$ and $42$ .

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Step 8.3.3.1

Factor $7$ out of $7 \sqrt{42}$ .

$\sec (x) = \frac{7 (\sqrt{42})}{42}$

Step 8.3.3.2

Cancel the common factors.

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Step 8.3.3.2.1

Factor $7$ out of $42$ .

$\sec (x) = \frac{7 \sqrt{42}}{7 \cdot 6}$

Step 8.3.3.2.2

Cancel the common factor.

$\sec (x) = \frac{7 \sqrt{42}}{7 \cdot 6}$

Step 8.3.3.2.3

Rewrite the expression.

$\sec (x) = \frac{\sqrt{42}}{6}$

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.

$\csc (x) = \frac{7}{\sqrt{7}}$

Step 9.3

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

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Step 9.3.1

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

$\csc (x) = \frac{7}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 9.3.2

Combine and simplify the denominator.

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Step 9.3.2.1

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

$\csc (x) = \frac{7 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 9.3.2.2

Raise $\sqrt{7}$ to the power of $1$ .

$\csc (x) = \frac{7 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 9.3.2.3

Raise $\sqrt{7}$ to the power of $1$ .

$\csc (x) = \frac{7 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 9.3.2.4

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

$\csc (x) = \frac{7 \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 9.3.2.5

Add $1$ and $1$ .

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

Step 9.3.2.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 9.3.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

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

Step 9.3.2.6.2

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

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

Step 9.3.2.6.3

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

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

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.

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

Step 9.3.2.6.4.2

Rewrite the expression.

$\csc (x) = \frac{7 \sqrt{7}}{7}$

Step 9.3.2.6.5

Evaluate the exponent.

$\csc (x) = \frac{7 \sqrt{7}}{7}$

Step 9.3.3

Cancel the common factor of $7$ .

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Step 9.3.3.1

Cancel the common factor.

$\csc (x) = \frac{7 \sqrt{7}}{7}$

Step 9.3.3.2

Divide $\sqrt{7}$ by $1$ .

$\csc (x) = \sqrt{7}$

Step 10

This is the solution to each trig value.

$\sin (x) = \frac{\sqrt{7}}{7}$

$\cos (x) = \frac{\sqrt{42}}{7}$

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

$\cot (x) = \sqrt{6}$

$\sec (x) = \frac{\sqrt{42}}{6}$

$\csc (x) = \sqrt{7}$