Trigonometry Examples

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

Step 1

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

$\cos (x) = \frac{adjacent}{hypotenuse}$

Step 2

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

$Opposite = \sqrt{{hypotenuse}^{2} - {adjacent}^{2}}$

Step 3

Replace the known values in the equation.

$Opposite = \sqrt{{(3)}^{2} - {(2 \sqrt{2})}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Raise $3$ to the power of $2$ .

Opposite $= \sqrt{9 - {(2 \sqrt{2})}^{2}}$

Step 4.2

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

Opposite $= \sqrt{9 - (2^{2} {\sqrt{2}}^{2})}$

Step 4.3

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

Opposite $= \sqrt{9 - (4 {\sqrt{2}}^{2})}$

Step 4.4

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

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Step 4.4.1

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

Opposite $= \sqrt{9 - (4 {(2^{\frac{1}{2}})}^{2})}$

Step 4.4.2

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

Opposite $= \sqrt{9 - (4 \cdot 2^{\frac{1}{2} \cdot 2})}$

Step 4.4.3

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

Opposite $= \sqrt{9 - (4 \cdot 2^{\frac{2}{2}})}$

Step 4.4.4

Cancel the common factor of $2$ .

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Step 4.4.4.1

Cancel the common factor.

Opposite $= \sqrt{9 - (4 \cdot 2^{\frac{2}{2}})}$

Step 4.4.4.2

Rewrite the expression.

Opposite $= \sqrt{9 - (4 \cdot 2)}$

Step 4.4.5

Evaluate the exponent.

Opposite $= \sqrt{9 - (4 \cdot 2)}$

Step 4.5

Multiply $- (4 \cdot 2)$ .

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Step 4.5.1

Multiply $4$ by $2$ .

Opposite $= \sqrt{9 - 1 \cdot 8}$

Step 4.5.2

Multiply $- 1$ by $8$ .

Opposite $= \sqrt{9 - 8}$

Step 4.6

Subtract $8$ from $9$ .

Opposite $= \sqrt{1}$

Step 4.7

Any root of $1$ is $1$ .

Opposite $= 1$

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.

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

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.

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

Step 6.3

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

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Step 6.3.1

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

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

Step 6.3.2

Combine and simplify the denominator.

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Step 6.3.2.1

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

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

Step 6.3.2.2

Move $\sqrt{2}$ .

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

Step 6.3.2.3

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

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

Step 6.3.2.4

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

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

Step 6.3.2.5

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

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

Step 6.3.2.6

Add $1$ and $1$ .

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

Step 6.3.2.7

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

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Step 6.3.2.7.1

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

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

Step 6.3.2.7.2

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

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

Step 6.3.2.7.3

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

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

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.

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

Step 6.3.2.7.4.2

Rewrite the expression.

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

Step 6.3.2.7.5

Evaluate the exponent.

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

Step 6.3.3

Multiply $2$ by $2$ .

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

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.

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

Step 7.3

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

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

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.

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

Step 8.3

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

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Step 8.3.1

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

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

Step 8.3.2

Combine and simplify the denominator.

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Step 8.3.2.1

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

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

Step 8.3.2.2

Move $\sqrt{2}$ .

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

Step 8.3.2.3

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

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

Step 8.3.2.4

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

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

Step 8.3.2.5

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

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

Step 8.3.2.6

Add $1$ and $1$ .

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

Step 8.3.2.7

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

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Step 8.3.2.7.1

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

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

Step 8.3.2.7.2

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

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

Step 8.3.2.7.3

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

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

Step 8.3.2.7.4

Cancel the common factor of $2$ .

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Step 8.3.2.7.4.1

Cancel the common factor.

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

Step 8.3.2.7.4.2

Rewrite the expression.

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

Step 8.3.2.7.5

Evaluate the exponent.

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

Step 8.3.3

Multiply $2$ by $2$ .

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

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.

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

Step 9.3

Divide $3$ by $1$ .

$\csc (x) = 3$

Step 10

This is the solution to each trig value.

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

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

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

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

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

$\csc (x) = 3$