Trigonometry Examples

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

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.

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

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.

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

Replace the known values in the equation.

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

Simplify inside the radical.

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Apply the product rule to $2 \sqrt{2}$ .

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

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

Hypotenuse $= \sqrt{4 {\sqrt{2}}^{2} + {(1)}^{2}}$

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

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

Hypotenuse $= \sqrt{4 {(2^{\frac{1}{2}})}^{2} + {(1)}^{2}}$

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

Hypotenuse $= \sqrt{4 \cdot 2^{\frac{1}{2} \cdot 2} + {(1)}^{2}}$

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

Hypotenuse $= \sqrt{4 \cdot 2^{\frac{2}{2}} + {(1)}^{2}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

Hypotenuse $= \sqrt{4 \cdot 2^{\frac{2}{2}} + {(1)}^{2}}$

Rewrite the expression.

Hypotenuse $= \sqrt{4 \cdot 2 + {(1)}^{2}}$

Evaluate the exponent.

Hypotenuse $= \sqrt{4 \cdot 2 + {(1)}^{2}}$

Multiply $4$ by $2$ .

Hypotenuse $= \sqrt{8 + {(1)}^{2}}$

One to any power is one.

Hypotenuse $= \sqrt{8 + 1}$

Add $8$ and $1$ .

Hypotenuse $= \sqrt{9}$

Rewrite $9$ as $3^{2}$ .

Hypotenuse $= \sqrt{3^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

Hypotenuse $= 3$

Find the value of sine.

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Use the definition of sine to find the value of $\sin (x)$ .

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

Substitute in the known values.

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

Find the value of cosine.

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

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

Substitute in the known values.

$\cos (x) = \frac{1}{3}$

Find the value of cotangent.

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

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

Substitute in the known values.

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

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

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

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

Combine and simplify the denominator.

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

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

Move $\sqrt{2}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $2$ by $2$ .

$\cot (x) = \frac{\sqrt{2}}{4}$

Find the value of secant.

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

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

Substitute in the known values.

$\sec (x) = \frac{3}{1}$

Divide $3$ by $1$ .

$\sec (x) = 3$

Find the value of cosecant.

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

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

Substitute in the known values.

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

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

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

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

Combine and simplify the denominator.

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

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

Move $\sqrt{2}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $2$ by $2$ .

$\csc (x) = \frac{3 \sqrt{2}}{4}$

This is the solution to each trig value.

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

$\cos (x) = \frac{1}{3}$

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

$\cot (x) = \frac{\sqrt{2}}{4}$

$\sec (x) = 3$

$\csc (x) = \frac{3 \sqrt{2}}{4}$