Precalculus Examples

$\cos (2 θ) = - \frac{\sqrt{3}}{2}$

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.

$\cos (2 θ) = \frac{adjacent}{hypotenuse}$

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.

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

Step 4

Simplify inside the radical.

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Step 4.1

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

Opposite $= \sqrt{4 - {(- \sqrt{3})}^{2}}$

Step 4.2

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

Opposite $= \sqrt{4 - ({(- 1)}^{2} {\sqrt{3}}^{2})}$

Step 4.3

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 4.3.1

Move ${(- 1)}^{2}$ .

Opposite $= \sqrt{4 + {(- 1)}^{2} \cdot (- 1 {\sqrt{3}}^{2})}$

Step 4.3.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 4.3.2.1

Raise $- 1$ to the power of $1$ .

Opposite $= \sqrt{4 + {(- 1)}^{2} \cdot ((- 1) {\sqrt{3}}^{2})}$

Step 4.3.2.2

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

Opposite $= \sqrt{4 + {(- 1)}^{2 + 1} {\sqrt{3}}^{2}}$

Step 4.3.3

Add $2$ and $1$ .

Opposite $= \sqrt{4 + {(- 1)}^{3} {\sqrt{3}}^{2}}$

Step 4.4

Raise $- 1$ to the power of $3$ .

Opposite $= \sqrt{4 - {\sqrt{3}}^{2}}$

Step 4.5

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

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Step 4.5.1

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

Opposite $= \sqrt{4 - {(3^{\frac{1}{2}})}^{2}}$

Step 4.5.2

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

Opposite $= \sqrt{4 - 3^{\frac{1}{2} \cdot 2}}$

Step 4.5.3

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

Opposite $= \sqrt{4 - 3^{\frac{2}{2}}}$

Step 4.5.4

Cancel the common factor of $2$ .

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Step 4.5.4.1

Cancel the common factor.

Opposite $= \sqrt{4 - 3^{\frac{2}{2}}}$

Step 4.5.4.2

Rewrite the expression.

Opposite $= \sqrt{4 - 3}$

Step 4.5.5

Evaluate the exponent.

Opposite $= \sqrt{4 - 1 \cdot 3}$

Step 4.6

Multiply $- 1$ by $3$ .

Opposite $= \sqrt{4 - 3}$

Step 4.7

Subtract $3$ from $4$ .

Opposite $= \sqrt{1}$

Step 4.8

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 $\sin (2 θ)$ .

$\sin (2 θ) = \frac{opp}{hyp}$

Step 5.2

Substitute in the known values.

$\sin (2 θ) = \frac{1}{2}$

Step 6

Find the value of tangent.

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Step 6.1

Use the definition of tangent to find the value of $\tan (2 θ)$ .

$\tan (2 θ) = \frac{opp}{adj}$

Step 6.2

Substitute in the known values.

$\tan (2 θ) = \frac{1}{- \sqrt{3}}$

Step 6.3

Simplify the value of $\tan (2 θ)$ .

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Step 6.3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 6.3.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$\tan (2 θ) = \frac{- 1 \cdot - 1}{- \sqrt{3}}$

Step 6.3.1.2

Move the negative in front of the fraction.

$\tan (2 θ) = - \frac{1}{\sqrt{3}}$

Step 6.3.2

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

$\tan (2 θ) = - (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 6.3.3

Combine and simplify the denominator.

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Step 6.3.3.1

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

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

Step 6.3.3.2

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

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

Step 6.3.3.3

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

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

Step 6.3.3.4

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

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

Step 6.3.3.5

Add $1$ and $1$ .

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

Step 6.3.3.6

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

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Step 6.3.3.6.1

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

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

Step 6.3.3.6.2

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

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

Step 6.3.3.6.3

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

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

Step 6.3.3.6.4

Cancel the common factor of $2$ .

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Step 6.3.3.6.4.1

Cancel the common factor.

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

Step 6.3.3.6.4.2

Rewrite the expression.

$\tan (2 θ) = - \frac{\sqrt{3}}{3}$

Step 6.3.3.6.5

Evaluate the exponent.

$\tan (2 θ) = - \frac{\sqrt{3}}{3}$

Step 7

Find the value of cotangent.

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Step 7.1

Use the definition of cotangent to find the value of $\cot (2 θ)$ .

$\cot (2 θ) = \frac{adj}{opp}$

Step 7.2

Substitute in the known values.

$\cot (2 θ) = \frac{- \sqrt{3}}{1}$

Step 7.3

Divide $- \sqrt{3}$ by $1$ .

$\cot (2 θ) = - \sqrt{3}$

Step 8

Find the value of secant.

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Step 8.1

Use the definition of secant to find the value of $\sec (2 θ)$ .

$\sec (2 θ) = \frac{hyp}{adj}$

Step 8.2

Substitute in the known values.

$\sec (2 θ) = \frac{2}{- \sqrt{3}}$

Step 8.3

Simplify the value of $\sec (2 θ)$ .

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Step 8.3.1

Move the negative in front of the fraction.

$\sec (2 θ) = - \frac{2}{\sqrt{3}}$

Step 8.3.2

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

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

Step 8.3.3

Combine and simplify the denominator.

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Step 8.3.3.1

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

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

Step 8.3.3.2

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

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

Step 8.3.3.3

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

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

Step 8.3.3.4

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

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

Step 8.3.3.5

Add $1$ and $1$ .

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

Step 8.3.3.6

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

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Step 8.3.3.6.1

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

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

Step 8.3.3.6.2

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

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

Step 8.3.3.6.3

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

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

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.

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

Step 8.3.3.6.4.2

Rewrite the expression.

$\sec (2 θ) = - \frac{2 \sqrt{3}}{3}$

Step 8.3.3.6.5

Evaluate the exponent.

$\sec (2 θ) = - \frac{2 \sqrt{3}}{3}$

Step 9

Find the value of cosecant.

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Step 9.1

Use the definition of cosecant to find the value of $\csc (2 θ)$ .

$\csc (2 θ) = \frac{hyp}{opp}$

Step 9.2

Substitute in the known values.

$\csc (2 θ) = \frac{2}{1}$

Step 9.3

Divide $2$ by $1$ .

$\csc (2 θ) = 2$

Step 10

This is the solution to each trig value.

$\sin (2 θ) = \frac{1}{2}$

$\cos (2 θ) = - \frac{\sqrt{3}}{2}$

$\tan (2 θ) = - \frac{\sqrt{3}}{3}$

$\cot (2 θ) = - \sqrt{3}$

$\sec (2 θ) = - \frac{2 \sqrt{3}}{3}$

$\csc (2 θ) = 2$