Precalculus Examples

$5 \tan (x) = 5 \sqrt{3}$

Step 1

Use the definition of tangent to find the known sides of the unit circle right triangle. The quadrant determines the sign on each of the values.

$\tan (x) = \frac{opposite}{adjacent}$

Step 2

Find the hypotenuse of the unit circle triangle. Since the opposite and adjacent sides are known, use the Pythagorean theorem to find the remaining side.

$Hypotenuse = \sqrt{{opposite}^{2} + {adjacent}^{2}}$

Step 3

Replace the known values in the equation.

$Hypotenuse = \sqrt{{(5 \sqrt{3})}^{2} + {(1)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Apply the product rule to $5 \sqrt{3}$ .

Hypotenuse $= \sqrt{5^{2} {\sqrt{3}}^{2} + {(1)}^{2}}$

Step 4.2

Raise $5$ to the power of $2$ .

Hypotenuse $= \sqrt{25 {\sqrt{3}}^{2} + {(1)}^{2}}$

Step 4.3

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

Hypotenuse $= \sqrt{25 {(3^{\frac{1}{2}})}^{2} + {(1)}^{2}}$

Step 4.3.2

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

Hypotenuse $= \sqrt{25 \cdot 3^{\frac{1}{2} \cdot 2} + {(1)}^{2}}$

Step 4.3.3

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

Hypotenuse $= \sqrt{25 \cdot 3^{\frac{2}{2}} + {(1)}^{2}}$

Step 4.3.4

Cancel the common factor of $2$ .

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Step 4.3.4.1

Cancel the common factor.

Hypotenuse $= \sqrt{25 \cdot 3^{\frac{2}{2}} + {(1)}^{2}}$

Step 4.3.4.2

Rewrite the expression.

Hypotenuse $= \sqrt{25 \cdot 3 + {(1)}^{2}}$

Step 4.3.5

Evaluate the exponent.

Hypotenuse $= \sqrt{25 \cdot 3 + {(1)}^{2}}$

Step 4.4

Multiply $25$ by $3$ .

Hypotenuse $= \sqrt{75 + {(1)}^{2}}$

Step 4.5

One to any power is one.

Hypotenuse $= \sqrt{75 + 1}$

Step 4.6

Add $75$ and $1$ .

Hypotenuse $= \sqrt{76}$

Step 4.7

Rewrite $76$ as $2^{2} \cdot 19$ .

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Step 4.7.1

Factor $4$ out of $76$ .

Hypotenuse $= \sqrt{4 (19)}$

Step 4.7.2

Rewrite $4$ as $2^{2}$ .

Hypotenuse $= \sqrt{2^{2} \cdot 19}$

Step 4.8

Pull terms out from under the radical.

Hypotenuse $= 2 \sqrt{19}$

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.

$\sin (x) = \frac{5 \sqrt{3}}{2 \sqrt{19}}$

Step 5.3

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

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Step 5.3.1

Multiply $\frac{5 \sqrt{3}}{2 \sqrt{19}}$ by $\frac{\sqrt{19}}{\sqrt{19}}$ .

$\sin (x) = \frac{5 \sqrt{3}}{2 \sqrt{19}} \cdot \frac{\sqrt{19}}{\sqrt{19}}$

Step 5.3.2

Combine and simplify the denominator.

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Step 5.3.2.1

Multiply $\frac{5 \sqrt{3}}{2 \sqrt{19}}$ by $\frac{\sqrt{19}}{\sqrt{19}}$ .

$\sin (x) = \frac{5 \sqrt{3} \sqrt{19}}{2 \sqrt{19} \sqrt{19}}$

Step 5.3.2.2

Move $\sqrt{19}$ .

$\sin (x) = \frac{5 \sqrt{3} \sqrt{19}}{2 (\sqrt{19} \sqrt{19})}$

Step 5.3.2.3

Raise $\sqrt{19}$ to the power of $1$ .

$\sin (x) = \frac{5 \sqrt{3} \sqrt{19}}{2 (\sqrt{19} \sqrt{19})}$

Step 5.3.2.4

Raise $\sqrt{19}$ to the power of $1$ .

$\sin (x) = \frac{5 \sqrt{3} \sqrt{19}}{2 (\sqrt{19} \sqrt{19})}$

Step 5.3.2.5

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

$\sin (x) = \frac{5 \sqrt{3} \sqrt{19}}{2 {\sqrt{19}}^{1 + 1}}$

Step 5.3.2.6

Add $1$ and $1$ .

$\sin (x) = \frac{5 \sqrt{3} \sqrt{19}}{2 {\sqrt{19}}^{2}}$

Step 5.3.2.7

Rewrite ${\sqrt{19}}^{2}$ as $19$ .

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Step 5.3.2.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{19}$ as $19^{\frac{1}{2}}$ .

$\sin (x) = \frac{5 \sqrt{3} \sqrt{19}}{2 {(19^{\frac{1}{2}})}^{2}}$

Step 5.3.2.7.2

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

$\sin (x) = \frac{5 \sqrt{3} \sqrt{19}}{2 \cdot 19^{\frac{1}{2} \cdot 2}}$

Step 5.3.2.7.3

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

$\sin (x) = \frac{5 \sqrt{3} \sqrt{19}}{2 \cdot 19^{\frac{2}{2}}}$

Step 5.3.2.7.4

Cancel the common factor of $2$ .

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Step 5.3.2.7.4.1

Cancel the common factor.

$\sin (x) = \frac{5 \sqrt{3} \sqrt{19}}{2 \cdot 19^{\frac{2}{2}}}$

Step 5.3.2.7.4.2

Rewrite the expression.

$\sin (x) = \frac{5 \sqrt{3} \sqrt{19}}{2 \cdot 19}$

Step 5.3.2.7.5

Evaluate the exponent.

$\sin (x) = \frac{5 \sqrt{3} \sqrt{19}}{2 \cdot 19}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Combine using the product rule for radicals.

$\sin (x) = \frac{5 \sqrt{19 \cdot 3}}{2 \cdot 19}$

Step 5.3.3.2

Multiply $19$ by $3$ .

$\sin (x) = \frac{5 \sqrt{57}}{2 \cdot 19}$

Step 5.3.4

Multiply $2$ by $19$ .

$\sin (x) = \frac{5 \sqrt{57}}{38}$

Step 6

Find the value of cosine.

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Step 6.1

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

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

Step 6.2

Substitute in the known values.

$\cos (x) = \frac{1}{2 \sqrt{19}}$

Step 6.3

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

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Step 6.3.1

Multiply $\frac{1}{2 \sqrt{19}}$ by $\frac{\sqrt{19}}{\sqrt{19}}$ .

$\cos (x) = \frac{1}{2 \sqrt{19}} \cdot \frac{\sqrt{19}}{\sqrt{19}}$

Step 6.3.2

Combine and simplify the denominator.

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Step 6.3.2.1

Multiply $\frac{1}{2 \sqrt{19}}$ by $\frac{\sqrt{19}}{\sqrt{19}}$ .

$\cos (x) = \frac{\sqrt{19}}{2 \sqrt{19} \sqrt{19}}$

Step 6.3.2.2

Move $\sqrt{19}$ .

$\cos (x) = \frac{\sqrt{19}}{2 (\sqrt{19} \sqrt{19})}$

Step 6.3.2.3

Raise $\sqrt{19}$ to the power of $1$ .

$\cos (x) = \frac{\sqrt{19}}{2 (\sqrt{19} \sqrt{19})}$

Step 6.3.2.4

Raise $\sqrt{19}$ to the power of $1$ .

$\cos (x) = \frac{\sqrt{19}}{2 (\sqrt{19} \sqrt{19})}$

Step 6.3.2.5

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

$\cos (x) = \frac{\sqrt{19}}{2 {\sqrt{19}}^{1 + 1}}$

Step 6.3.2.6

Add $1$ and $1$ .

$\cos (x) = \frac{\sqrt{19}}{2 {\sqrt{19}}^{2}}$

Step 6.3.2.7

Rewrite ${\sqrt{19}}^{2}$ as $19$ .

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Step 6.3.2.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{19}$ as $19^{\frac{1}{2}}$ .

$\cos (x) = \frac{\sqrt{19}}{2 {(19^{\frac{1}{2}})}^{2}}$

Step 6.3.2.7.2

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

$\cos (x) = \frac{\sqrt{19}}{2 \cdot 19^{\frac{1}{2} \cdot 2}}$

Step 6.3.2.7.3

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

$\cos (x) = \frac{\sqrt{19}}{2 \cdot 19^{\frac{2}{2}}}$

Step 6.3.2.7.4

Cancel the common factor of $2$ .

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Step 6.3.2.7.4.1

Cancel the common factor.

$\cos (x) = \frac{\sqrt{19}}{2 \cdot 19^{\frac{2}{2}}}$

Step 6.3.2.7.4.2

Rewrite the expression.

$\cos (x) = \frac{\sqrt{19}}{2 \cdot 19}$

Step 6.3.2.7.5

Evaluate the exponent.

$\cos (x) = \frac{\sqrt{19}}{2 \cdot 19}$

Step 6.3.3

Multiply $2$ by $19$ .

$\cos (x) = \frac{\sqrt{19}}{38}$

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.

$\cot (x) = \frac{1}{5 \sqrt{3}}$

Step 7.3

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

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Step 7.3.1

Multiply $\frac{1}{5 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\cot (x) = \frac{1}{5 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 7.3.2

Combine and simplify the denominator.

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Step 7.3.2.1

Multiply $\frac{1}{5 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\cot (x) = \frac{\sqrt{3}}{5 \sqrt{3} \sqrt{3}}$

Step 7.3.2.2

Move $\sqrt{3}$ .

$\cot (x) = \frac{\sqrt{3}}{5 (\sqrt{3} \sqrt{3})}$

Step 7.3.2.3

Raise $\sqrt{3}$ to the power of $1$ .

$\cot (x) = \frac{\sqrt{3}}{5 (\sqrt{3} \sqrt{3})}$

Step 7.3.2.4

Raise $\sqrt{3}$ to the power of $1$ .

$\cot (x) = \frac{\sqrt{3}}{5 (\sqrt{3} \sqrt{3})}$

Step 7.3.2.5

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

$\cot (x) = \frac{\sqrt{3}}{5 {\sqrt{3}}^{1 + 1}}$

Step 7.3.2.6

Add $1$ and $1$ .

$\cot (x) = \frac{\sqrt{3}}{5 {\sqrt{3}}^{2}}$

Step 7.3.2.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 7.3.2.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\cot (x) = \frac{\sqrt{3}}{5 {(3^{\frac{1}{2}})}^{2}}$

Step 7.3.2.7.2

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

$\cot (x) = \frac{\sqrt{3}}{5 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 7.3.2.7.3

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

$\cot (x) = \frac{\sqrt{3}}{5 \cdot 3^{\frac{2}{2}}}$

Step 7.3.2.7.4

Cancel the common factor of $2$ .

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Step 7.3.2.7.4.1

Cancel the common factor.

$\cot (x) = \frac{\sqrt{3}}{5 \cdot 3^{\frac{2}{2}}}$

Step 7.3.2.7.4.2

Rewrite the expression.

$\cot (x) = \frac{\sqrt{3}}{5 \cdot 3}$

Step 7.3.2.7.5

Evaluate the exponent.

$\cot (x) = \frac{\sqrt{3}}{5 \cdot 3}$

Step 7.3.3

Multiply $5$ by $3$ .

$\cot (x) = \frac{\sqrt{3}}{15}$

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.

$\sec (x) = \frac{2 \sqrt{19}}{1}$

Step 8.3

Divide $2 \sqrt{19}$ by $1$ .

$\sec (x) = 2 \sqrt{19}$

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.

$\csc (x) = \frac{2 \sqrt{19}}{5 \sqrt{3}}$

Step 9.3

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

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Step 9.3.1

Multiply $\frac{2 \sqrt{19}}{5 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\csc (x) = \frac{2 \sqrt{19}}{5 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 9.3.2

Combine and simplify the denominator.

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Step 9.3.2.1

Multiply $\frac{2 \sqrt{19}}{5 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\csc (x) = \frac{2 \sqrt{19} \sqrt{3}}{5 \sqrt{3} \sqrt{3}}$

Step 9.3.2.2

Move $\sqrt{3}$ .

$\csc (x) = \frac{2 \sqrt{19} \sqrt{3}}{5 (\sqrt{3} \sqrt{3})}$

Step 9.3.2.3

Raise $\sqrt{3}$ to the power of $1$ .

$\csc (x) = \frac{2 \sqrt{19} \sqrt{3}}{5 (\sqrt{3} \sqrt{3})}$

Step 9.3.2.4

Raise $\sqrt{3}$ to the power of $1$ .

$\csc (x) = \frac{2 \sqrt{19} \sqrt{3}}{5 (\sqrt{3} \sqrt{3})}$

Step 9.3.2.5

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

$\csc (x) = \frac{2 \sqrt{19} \sqrt{3}}{5 {\sqrt{3}}^{1 + 1}}$

Step 9.3.2.6

Add $1$ and $1$ .

$\csc (x) = \frac{2 \sqrt{19} \sqrt{3}}{5 {\sqrt{3}}^{2}}$

Step 9.3.2.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 9.3.2.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\csc (x) = \frac{2 \sqrt{19} \sqrt{3}}{5 {(3^{\frac{1}{2}})}^{2}}$

Step 9.3.2.7.2

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

$\csc (x) = \frac{2 \sqrt{19} \sqrt{3}}{5 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 9.3.2.7.3

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

$\csc (x) = \frac{2 \sqrt{19} \sqrt{3}}{5 \cdot 3^{\frac{2}{2}}}$

Step 9.3.2.7.4

Cancel the common factor of $2$ .

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Step 9.3.2.7.4.1

Cancel the common factor.

$\csc (x) = \frac{2 \sqrt{19} \sqrt{3}}{5 \cdot 3^{\frac{2}{2}}}$

Step 9.3.2.7.4.2

Rewrite the expression.

$\csc (x) = \frac{2 \sqrt{19} \sqrt{3}}{5 \cdot 3}$

Step 9.3.2.7.5

Evaluate the exponent.

$\csc (x) = \frac{2 \sqrt{19} \sqrt{3}}{5 \cdot 3}$

Step 9.3.3

Simplify the numerator.

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Step 9.3.3.1

Combine using the product rule for radicals.

$\csc (x) = \frac{2 \sqrt{3 \cdot 19}}{5 \cdot 3}$

Step 9.3.3.2

Multiply $3$ by $19$ .

$\csc (x) = \frac{2 \sqrt{57}}{5 \cdot 3}$

Step 9.3.4

Multiply $5$ by $3$ .

$\csc (x) = \frac{2 \sqrt{57}}{15}$

Step 10

This is the solution to each trig value.

$\sin (x) = \frac{5 \sqrt{57}}{38}$

$\cos (x) = \frac{\sqrt{19}}{38}$

$\tan (x) = 5 \sqrt{3}$

$\cot (x) = \frac{\sqrt{3}}{15}$

$\sec (x) = 2 \sqrt{19}$

$\csc (x) = \frac{2 \sqrt{57}}{15}$