Trigonometry Examples

$\sin (x) = \frac{3}{4}$ , $\cos (x) = \frac{\sqrt{7}}{4}$

Step 1

Identify the known sides of the triangle using $\sin (x) = \frac{3}{4}$ .

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

Step 2

Identify the known sides of the triangle using $\cos (x) = \frac{\sqrt{7}}{4}$ .

$\cos (x) = \frac{adj}{hyp} = \frac{\sqrt{7}}{4}$

Step 3

Use $opp = 3$ and $adj = \sqrt{7}$ to find the value of the tangent function.

$\tan (x) = \frac{opp}{adj} = \frac{3}{\sqrt{7}}$

Step 4

Simplify the result.

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Step 4.1

Multiply $\frac{3}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\tan (x) = \frac{opp}{adj} = \frac{3}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 4.2

Combine and simplify the denominator.

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Step 4.2.1

Multiply $\frac{3}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\tan (x) = \frac{opp}{adj} = \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 4.2.2

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

$\tan (x) = \frac{opp}{adj} = \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 4.2.3

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

$\tan (x) = \frac{opp}{adj} = \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 4.2.4

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

$\tan (x) = \frac{opp}{adj} = \frac{3 \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 4.2.5

Add $1$ and $1$ .

$\tan (x) = \frac{opp}{adj} = \frac{3 \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 4.2.6

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

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Step 4.2.6.1

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

$\tan (x) = \frac{opp}{adj} = \frac{3 \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 4.2.6.2

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

$\tan (x) = \frac{opp}{adj} = \frac{3 \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 4.2.6.3

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

$\tan (x) = \frac{opp}{adj} = \frac{3 \sqrt{7}}{7^{\frac{2}{2}}}$

Step 4.2.6.4

Cancel the common factor of $2$ .

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Step 4.2.6.4.1

Cancel the common factor.

$\tan (x) = \frac{opp}{adj} = \frac{3 \sqrt{7}}{7^{\frac{2}{2}}}$

Step 4.2.6.4.2

Rewrite the expression.

$\tan (x) = \frac{opp}{adj} = \frac{3 \sqrt{7}}{7}$

Step 4.2.6.5

Evaluate the exponent.

$\tan (x) = \frac{opp}{adj} = \frac{3 \sqrt{7}}{7}$

Step 5

Use $hyp = 4$ and $adj = \sqrt{7}$ to find the value of the secant function.

$\sec (x) = \frac{1}{\cos (x)} = \frac{hyp}{adj} = \frac{4}{\sqrt{7}}$

Step 6

Simplify the result.

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Step 6.1

Multiply $\frac{4}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\sec (x) = \frac{1}{\cos (x)} = \frac{hyp}{adj} = \frac{4}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 6.2

Combine and simplify the denominator.

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Step 6.2.1

Multiply $\frac{4}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\sec (x) = \frac{1}{\cos (x)} = \frac{hyp}{adj} = \frac{4 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 6.2.2

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

$\sec (x) = \frac{1}{\cos (x)} = \frac{hyp}{adj} = \frac{4 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 6.2.3

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

$\sec (x) = \frac{1}{\cos (x)} = \frac{hyp}{adj} = \frac{4 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 6.2.4

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

$\sec (x) = \frac{1}{\cos (x)} = \frac{hyp}{adj} = \frac{4 \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 6.2.5

Add $1$ and $1$ .

$\sec (x) = \frac{1}{\cos (x)} = \frac{hyp}{adj} = \frac{4 \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 6.2.6

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

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Step 6.2.6.1

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

$\sec (x) = \frac{1}{\cos (x)} = \frac{hyp}{adj} = \frac{4 \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 6.2.6.2

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

$\sec (x) = \frac{1}{\cos (x)} = \frac{hyp}{adj} = \frac{4 \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 6.2.6.3

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

$\sec (x) = \frac{1}{\cos (x)} = \frac{hyp}{adj} = \frac{4 \sqrt{7}}{7^{\frac{2}{2}}}$

Step 6.2.6.4

Cancel the common factor of $2$ .

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Step 6.2.6.4.1

Cancel the common factor.

$\sec (x) = \frac{1}{\cos (x)} = \frac{hyp}{adj} = \frac{4 \sqrt{7}}{7^{\frac{2}{2}}}$

Step 6.2.6.4.2

Rewrite the expression.

$\sec (x) = \frac{1}{\cos (x)} = \frac{hyp}{adj} = \frac{4 \sqrt{7}}{7}$

Step 6.2.6.5

Evaluate the exponent.

$\sec (x) = \frac{1}{\cos (x)} = \frac{hyp}{adj} = \frac{4 \sqrt{7}}{7}$

Step 7

Use $opp = 3$ and $adj = \sqrt{7}$ to find the value of the cotangent function.

$\cot (x) = \frac{1}{\tan (x)} = \frac{adj}{opp} = \frac{\sqrt{7}}{3}$

Step 8

Use $opp = 3$ and $adj = \sqrt{7}$ to find the value of the cosecant function.

$\csc (x) = \frac{1}{\sin (x)} = \frac{hyp}{opp} = \frac{4}{3}$

Step 9

The trig functions found are as follows:

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

$\cos (x) = \frac{\sqrt{7}}{4}$

$\tan (x) = \frac{3 \sqrt{7}}{7}$

$\cot (x) = \frac{\sqrt{7}}{3}$

$\sec (x) = \frac{4 \sqrt{7}}{7}$

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