Trigonometry Examples

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

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.

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

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.

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

Step 4

Simplify inside the radical.

Tap for more steps...

Step 4.1

Negate $\sqrt{{(2 \sqrt{3})}^{2} - {(3)}^{2}}$ .

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

Step 4.2

Apply the product rule to $2 \sqrt{3}$ .

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

Step 4.3

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

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

Step 4.4

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

Tap for more steps...

Step 4.4.1

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

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

Step 4.4.2

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

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

Step 4.4.3

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

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

Step 4.4.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.4.4.1

Cancel the common factor.

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

Step 4.4.4.2

Rewrite the expression.

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

Step 4.4.5

Evaluate the exponent.

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

Step 4.5

Multiply $4$ by $3$ .

Opposite $= - \sqrt{12 - {(3)}^{2}}$

Step 4.6

Raise $3$ to the power of $2$ .

Opposite $= - \sqrt{12 - 1 \cdot 9}$

Step 4.7

Multiply $- 1$ by $9$ .

Opposite $= - \sqrt{12 - 9}$

Step 4.8

Subtract $9$ from $12$ .

Opposite $= - \sqrt{3}$

Step 5

Find the value of sine.

Tap for more steps...

Step 5.1

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

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

Step 5.2

Substitute in the known values.

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

Step 5.3

Simplify the value of $\sin (x)$ .

Tap for more steps...

Step 5.3.1

Cancel the common factor of $\sqrt{3}$ .

Tap for more steps...

Step 5.3.1.1

Cancel the common factor.

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

Step 5.3.1.2

Rewrite the expression.

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

Step 5.3.2

Move the negative in front of the fraction.

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

Step 6

Find the value of cosine.

Tap for more steps...

Step 6.1

Use the definition of cosine to find the value of $\cos (x)$ .

$\cos (x) = \frac{adj}{hyp}$

Step 6.2

Substitute in the known values.

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

Step 6.3

Simplify the value of $\cos (x)$ .

Tap for more steps...

Step 6.3.1

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

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

Step 6.3.2

Combine and simplify the denominator.

Tap for more steps...

Step 6.3.2.1

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

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

Step 6.3.2.2

Move $\sqrt{3}$ .

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

Step 6.3.2.3

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

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

Step 6.3.2.4

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

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

Step 6.3.2.5

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

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

Step 6.3.2.6

Add $1$ and $1$ .

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

Step 6.3.2.7

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

Tap for more steps...

Step 6.3.2.7.1

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

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

Step 6.3.2.7.2

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

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

Step 6.3.2.7.3

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

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

Step 6.3.2.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.2.7.4.1

Cancel the common factor.

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

Step 6.3.2.7.4.2

Rewrite the expression.

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

Step 6.3.2.7.5

Evaluate the exponent.

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

Step 6.3.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.3.3.1

Cancel the common factor.

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

Step 6.3.3.2

Rewrite the expression.

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

Step 7

Find the value of tangent.

Tap for more steps...

Step 7.1

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

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

Step 7.2

Substitute in the known values.

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

Step 7.3

Move the negative in front of the fraction.

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

Step 8

Find the value of cotangent.

Tap for more steps...

Step 8.1

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

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

Step 8.2

Substitute in the known values.

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

Step 8.3

Simplify the value of $\cot (x)$ .

Tap for more steps...

Step 8.3.1

Move the negative in front of the fraction.

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

Step 8.3.2

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

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

Step 8.3.3

Combine and simplify the denominator.

Tap for more steps...

Step 8.3.3.1

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

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

Step 8.3.3.2

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

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

Step 8.3.3.3

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

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

Step 8.3.3.4

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

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

Step 8.3.3.5

Add $1$ and $1$ .

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

Step 8.3.3.6

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

Tap for more steps...

Step 8.3.3.6.1

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

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

Step 8.3.3.6.2

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

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

Step 8.3.3.6.3

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

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

Step 8.3.3.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.3.3.6.4.1

Cancel the common factor.

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

Step 8.3.3.6.4.2

Rewrite the expression.

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

Step 8.3.3.6.5

Evaluate the exponent.

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

Step 8.3.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 8.3.4.1

Cancel the common factor.

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

Step 8.3.4.2

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

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

Step 9

Find the value of cosecant.

Tap for more steps...

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.

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

Step 9.3

Simplify the value of $\csc (x)$ .

Tap for more steps...

Step 9.3.1

Cancel the common factor of $\sqrt{3}$ .

Tap for more steps...

Step 9.3.1.1

Cancel the common factor.

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

Step 9.3.1.2

Rewrite the expression.

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

Step 9.3.1.3

Move the negative one from the denominator of $\frac{2}{- 1}$ .

$\csc (x) = - 1 \cdot 2$

Step 9.3.2

Multiply $- 1$ by $2$ .

$\csc (x) = - 2$

Step 10

This is the solution to each trig value.

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

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

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

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

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

$\csc (x) = - 2$