Find the Trig Value cos(2x)=-( square root of 3)/2

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

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.

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

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.

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

Step 4

Simplify inside the radical.

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

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

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

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

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

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

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

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

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

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

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

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

Add $2$ and $1$ .

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

Opposite $= \sqrt{4 - 3}$

Evaluate the exponent.

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

Multiply $- 1$ by $3$ .

Opposite $= \sqrt{4 - 3}$

Subtract $3$ from $4$ .

Opposite $= \sqrt{1}$

Any root of $1$ is $1$ .

Opposite $= 1$

Step 5

Find the value of sine.

Use the definition of sine to find the value of $\sin (2 x)$ .

$\sin (2 x) = \frac{opp}{hyp}$

Substitute in the known values.

$\sin (2 x) = \frac{1}{2}$

Step 6

Find the value of tangent.

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

$\tan (2 x) = \frac{opp}{adj}$

Substitute in the known values.

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

Simplify the value of $\tan (2 x)$ .

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

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

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

Move the negative in front of the fraction.

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 7

Find the value of cotangent.

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

$\cot (2 x) = \frac{adj}{opp}$

Substitute in the known values.

$\cot (2 x) = \frac{- \sqrt{3}}{1}$

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

$\cot (2 x) = - \sqrt{3}$

Step 8

Find the value of secant.

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

$\sec (2 x) = \frac{hyp}{adj}$

Substitute in the known values.

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

Simplify the value of $\sec (2 x)$ .

Move the negative in front of the fraction.

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 9

Find the value of cosecant.

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

$\csc (2 x) = \frac{hyp}{opp}$

Substitute in the known values.

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

Divide $2$ by $1$ .

$\csc (2 x) = 2$

Step 10

This is the solution to each trig value.

$\sin (2 x) = \frac{1}{2}$

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

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

$\cot (2 x) = - \sqrt{3}$

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

$\csc (2 x) = 2$

Find the Trig Value cos(2x)=-( square root of 3)/2

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