Find the Trig Value sec(30/2)=-2

$\sec (\frac{30}{2}) = - 2$

Step 1

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

$\sec (\frac{30}{2}) = \frac{hypotenuse}{adjacent}$

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.

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

Step 4

Simplify inside the radical.

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

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

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

Opposite $= \sqrt{4 - 1}$

Subtract $1$ from $4$ .

Opposite $= \sqrt{3}$

Step 5

Find the value of sine.

Use the definition of sine to find the value of $\sin (\frac{30}{2})$ .

$\sin (\frac{30}{2}) = \frac{opp}{hyp}$

Substitute in the known values.

$\sin (\frac{30}{2}) = \frac{\sqrt{3}}{2}$

Step 6

Find the value of cosine.

Use the definition of cosine to find the value of $\cos (\frac{30}{2})$ .

$\cos (\frac{30}{2}) = \frac{adj}{hyp}$

Substitute in the known values.

$\cos (\frac{30}{2}) = \frac{- 1}{2}$

Move the negative in front of the fraction.

$\cos (\frac{30}{2}) = - \frac{1}{2}$

Step 7

Find the value of tangent.

Use the definition of tangent to find the value of $\tan (\frac{30}{2})$ .

$\tan (\frac{30}{2}) = \frac{opp}{adj}$

Substitute in the known values.

$\tan (\frac{30}{2}) = \frac{\sqrt{3}}{- 1}$

Simplify the value of $\tan (\frac{30}{2})$ .

Move the negative one from the denominator of $\frac{\sqrt{3}}{- 1}$ .

$\tan (\frac{30}{2}) = - 1 \cdot \sqrt{3}$

Rewrite $- 1 \cdot \sqrt{3}$ as $- \sqrt{3}$ .

$\tan (\frac{30}{2}) = - \sqrt{3}$

Step 8

Find the value of cotangent.

Use the definition of cotangent to find the value of $\cot (\frac{30}{2})$ .

$\cot (\frac{30}{2}) = \frac{adj}{opp}$

Substitute in the known values.

$\cot (\frac{30}{2}) = \frac{- 1}{\sqrt{3}}$

Simplify the value of $\cot (\frac{30}{2})$ .

Move the negative in front of the fraction.

$\cot (\frac{30}{2}) = - \frac{1}{\sqrt{3}}$

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

$\cot (\frac{30}{2}) = - (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Combine and simplify the denominator.

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

$\cot (\frac{30}{2}) = - \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

$\cot (\frac{30}{2}) = - \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

$\cot (\frac{30}{2}) = - \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

$\cot (\frac{30}{2}) = - \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$\cot (\frac{30}{2}) = - \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

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

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

$\cot (\frac{30}{2}) = - \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

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

$\cot (\frac{30}{2}) = - \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

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

$\cot (\frac{30}{2}) = - \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\cot (\frac{30}{2}) = - \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Rewrite the expression.

$\cot (\frac{30}{2}) = - \frac{\sqrt{3}}{3}$

Evaluate the exponent.

$\cot (\frac{30}{2}) = - \frac{\sqrt{3}}{3}$

Step 9

Find the value of cosecant.

Use the definition of cosecant to find the value of $\csc (\frac{30}{2})$ .

$\csc (\frac{30}{2}) = \frac{hyp}{opp}$

Substitute in the known values.

$\csc (\frac{30}{2}) = \frac{2}{\sqrt{3}}$

Simplify the value of $\csc (\frac{30}{2})$ .

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

$\csc (\frac{30}{2}) = \frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Combine and simplify the denominator.

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

$\csc (\frac{30}{2}) = \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

$\csc (\frac{30}{2}) = \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

$\csc (\frac{30}{2}) = \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

$\csc (\frac{30}{2}) = \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$\csc (\frac{30}{2}) = \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}$

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

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

$\csc (\frac{30}{2}) = \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

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

$\csc (\frac{30}{2}) = \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

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

$\csc (\frac{30}{2}) = \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\csc (\frac{30}{2}) = \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Rewrite the expression.

$\csc (\frac{30}{2}) = \frac{2 \sqrt{3}}{3}$

Evaluate the exponent.

$\csc (\frac{30}{2}) = \frac{2 \sqrt{3}}{3}$

Step 10

This is the solution to each trig value.

$\sin (\frac{30}{2}) = \frac{\sqrt{3}}{2}$

$\cos (\frac{30}{2}) = - \frac{1}{2}$

$\tan (\frac{30}{2}) = - \sqrt{3}$

$\cot (\frac{30}{2}) = - \frac{\sqrt{3}}{3}$

$\sec (\frac{30}{2}) = - 2$

$\csc (\frac{30}{2}) = \frac{2 \sqrt{3}}{3}$

Find the Trig Value sec(30/2)=-2

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