Trigonometry Examples

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

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

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}

$\sin (x) = \frac{opposite}{hypotenuse}$

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}}

$Adjacent = - \sqrt{{hypotenuse}^{2} - {opposite}^{2}}$

Step 3

Replace the known values in the equation.

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

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

Step 4

Simplify inside the radical.

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Step 4.1

Negate

\sqrt{{(2)}^{2} - {(\sqrt{2})}^{2}}

$\sqrt{{(2)}^{2} - {(\sqrt{2})}^{2}}$ .

Adjacent

= - \sqrt{{(2)}^{2} - {(\sqrt{2})}^{2}}

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

Step 4.2

Raise

2

$2$ to the power of

2

$2$ .

Adjacent

= - \sqrt{4 - {(\sqrt{2})}^{2}}

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

Step 4.3

Rewrite

{\sqrt{2}}^{2}

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

2

$2$ .

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Step 4.3.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{2}

$\sqrt{2}$ as

2^{\frac{1}{2}}

$2^{\frac{1}{2}}$ .

Adjacent

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

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

Step 4.3.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

Adjacent

= - \sqrt{4 - 2^{\frac{1}{2} \cdot 2}}

$= - \sqrt{4 - 2^{\frac{1}{2} \cdot 2}}$

Step 4.3.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

Adjacent

= - \sqrt{4 - 2^{\frac{2}{2}}}

$= - \sqrt{4 - 2^{\frac{2}{2}}}$

Step 4.3.4

Cancel the common factor of

2

$2$ .

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Step 4.3.4.1

Cancel the common factor.

Adjacent

$= - \sqrt{4 - 2^{\frac{2}{2}}}$

Step 4.3.4.2

Rewrite the expression.

Adjacent

$= - \sqrt{4 - 2}$

Adjacent

$= - \sqrt{4 - 2}$

Step 4.3.5

Evaluate the exponent.

Adjacent

$= - \sqrt{4 - 1 \cdot 2}$

Adjacent

$= - \sqrt{4 - 1 \cdot 2}$

Step 4.4

Multiply

$- 1$ by

$2$ .

Adjacent

$= - \sqrt{4 - 2}$

Step 4.5

Subtract

$2$ from

$4$ .

Adjacent

$= - \sqrt{2}$

Adjacent

$= - \sqrt{2}$

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.

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

Step 5.3

Move the negative in front of the fraction.

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

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.

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

Step 6.3

Simplify the value of

$\tan (x)$ .

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Step 6.3.1

Cancel the common factor of

$\sqrt{2}$ .

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Step 6.3.1.1

Cancel the common factor.

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

Step 6.3.1.2

Rewrite the expression.

$\tan (x) = \frac{1}{- 1}$

Step 6.3.1.3

Move the negative one from the denominator of

$\frac{1}{- 1}$ .

$\tan (x) = - 1 \cdot 1$

Step 6.3.2

Multiply

$- 1$ by

$1$ .

$\tan (x) = - 1$

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.

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

Step 7.3

Cancel the common factor of

$\sqrt{2}$ .

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Step 7.3.1

Cancel the common factor.

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

Step 7.3.2

Divide

$- 1$ by

$1$ .

$\cot (x) = - 1$

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.

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

Step 8.3

Simplify the value of

$\sec (x)$ .

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Step 8.3.1

Move the negative in front of the fraction.

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

Step 8.3.2

Multiply

$\frac{2}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 8.3.3

Combine and simplify the denominator.

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Step 8.3.3.1

Multiply

$\frac{2}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 8.3.3.2

Raise

$\sqrt{2}$ to the power of

$1$ .

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

Step 8.3.3.3

Raise

$\sqrt{2}$ to the power of

$1$ .

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

Step 8.3.3.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 8.3.3.5

Add

$1$ and

$1$ .

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

Step 8.3.3.6

Rewrite

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

$2$ .

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Step 8.3.3.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

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

Step 8.3.3.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 8.3.3.6.3

Combine

$\frac{1}{2}$ and

$2$ .

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

Step 8.3.3.6.4

Cancel the common factor of

$2$ .

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Step 8.3.3.6.4.1

Cancel the common factor.

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

Step 8.3.3.6.4.2

Rewrite the expression.

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

Step 8.3.3.6.5

Evaluate the exponent.

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

Step 8.3.4

Cancel the common factor of

$2$ .

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Step 8.3.4.1

Cancel the common factor.

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

Step 8.3.4.2

Divide

$\sqrt{2}$ by

$1$ .

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

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.

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

Step 9.3

Simplify the value of

$\csc (x)$ .

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Step 9.3.1

Multiply

$\frac{2}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 9.3.2

Combine and simplify the denominator.

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Step 9.3.2.1

Multiply

$\frac{2}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 9.3.2.2

Raise

$\sqrt{2}$ to the power of

$1$ .

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

Step 9.3.2.3

Raise

$\sqrt{2}$ to the power of

$1$ .

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

Step 9.3.2.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 9.3.2.5

Add

$1$ and

$1$ .

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

Step 9.3.2.6

Rewrite

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

$2$ .

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Step 9.3.2.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

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

Step 9.3.2.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 9.3.2.6.3

Combine

$\frac{1}{2}$ and

$2$ .

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

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.

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

Step 9.3.2.6.4.2

Rewrite the expression.

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

Step 9.3.2.6.5

Evaluate the exponent.

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

Step 9.3.3

Cancel the common factor of

$2$ .

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Step 9.3.3.1

Cancel the common factor.

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

Step 9.3.3.2

Divide

$\sqrt{2}$ by

$1$ .

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

Step 10

This is the solution to each trig value.

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

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

$\tan (x) = - 1$

$\cot (x) = - 1$

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

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

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