Trigonometry Examples

sin (θ) = \frac{1}{2}

$\sin (θ) = \frac{1}{2}$ ,

sec (θ)

$\sec (θ)$

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 (θ) = \frac{opposite}{hypotenuse}

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

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} - {(1)}^{2}}

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

Step 4

Simplify inside the radical.

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Step 4.1

Raise

2

$2$ to the power of

2

$2$ .

Adjacent

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

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

Step 4.2

One to any power is one.

Adjacent

= \sqrt{4 - 1 \cdot 1}

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

Step 4.3

Multiply

- 1

$- 1$ by

1

$1$ .

Adjacent

= \sqrt{4 - 1}

$= \sqrt{4 - 1}$

Step 4.4

Subtract

1

$1$ from

4

$4$ .

Adjacent

= \sqrt{3}

$= \sqrt{3}$

Adjacent

= \sqrt{3}

$= \sqrt{3}$

Step 5

Use the definition of secant to find the value of

sec (θ)

$\sec (θ)$ .

sec (θ) = \frac{hypotenuse}{adjacent}

$\sec (θ) = \frac{hypotenuse}{adjacent}$

Step 6

Substitute in the known values.

sec (θ) = \frac{2}{\sqrt{3}}

$\sec (θ) = \frac{2}{\sqrt{3}}$

Step 7

Simplify the right side.

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Step 7.1

Multiply

\frac{2}{\sqrt{3}}

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

\frac{\sqrt{3}}{\sqrt{3}}

$\frac{\sqrt{3}}{\sqrt{3}}$ .

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

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

Step 7.2

Combine and simplify the denominator.

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Step 7.2.1

Multiply

\frac{2}{\sqrt{3}}

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

\frac{\sqrt{3}}{\sqrt{3}}

$\frac{\sqrt{3}}{\sqrt{3}}$ .

sec (θ) = \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}

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

Step 7.2.2

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

sec (θ) = \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}

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

Step 7.2.3

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

sec (θ) = \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}

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

Step 7.2.4

Use the power rule

a^{m} a^{n} = a^{m + n}

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

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

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

Step 7.2.5

Add

1

$1$ and

1

$1$ .

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

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

Step 7.2.6

Rewrite

{\sqrt{3}}^{2}

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

3

$3$ .

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Step 7.2.6.1

Use

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

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

\sqrt{3}

$\sqrt{3}$ as

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

$3^{\frac{1}{2}}$ .

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

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

Step 7.2.6.2

Apply the power rule and multiply exponents,

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

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

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

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

Step 7.2.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 7.2.6.4

Cancel the common factor of

2

$2$ .

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Step 7.2.6.4.1

Cancel the common factor.

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

Step 7.2.6.4.2

Rewrite the expression.

$\sec (θ) = \frac{2 \sqrt{3}}{3}$

Step 7.2.6.5

Evaluate the exponent.

$\sec (θ) = \frac{2 \sqrt{3}}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\sec (θ) = \frac{2 \sqrt{3}}{3}$

Decimal Form:

$\sec (θ) = 1.15470053 \dots$