Algebra Examples

$\cos (x) = - \frac{3}{\sqrt{10}}$

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.

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

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.

$Opposite = - \sqrt{{hypotenuse}^{2} - {adjacent}^{2}}$

Replace the known values in the equation.

$Opposite = - \sqrt{{(\sqrt{10})}^{2} - {(- 3)}^{2}}$

Simplify inside the radical.

Tap for more steps...

Negate $\sqrt{{(\sqrt{10})}^{2} - {(- 3)}^{2}}$ .

Opposite $= - \sqrt{{(\sqrt{10})}^{2} - {(- 3)}^{2}}$

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Tap for more steps...

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

Opposite $= - \sqrt{{(10^{\frac{1}{2}})}^{2} - {(- 3)}^{2}}$

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

Opposite $= - \sqrt{10^{\frac{1}{2} \cdot 2} - {(- 3)}^{2}}$

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

Opposite $= - \sqrt{10^{\frac{2}{2}} - {(- 3)}^{2}}$

Cancel the common factor of $2$ .

Tap for more steps...

Cancel the common factor.

Opposite $= - \sqrt{10^{\frac{2}{2}} - {(- 3)}^{2}}$

Rewrite the expression.

Opposite $= - \sqrt{10 - {(- 3)}^{2}}$

Evaluate the exponent.

Opposite $= - \sqrt{10 - {(- 3)}^{2}}$

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

Opposite $= - \sqrt{10 - 1 \cdot 9}$

Multiply $- 1$ by $9$ .

Opposite $= - \sqrt{10 - 9}$

Subtract $9$ from $10$ .

Opposite $= - \sqrt{1}$

Any root of $1$ is $1$ .

Opposite $= - 1 \cdot 1$

Multiply $- 1$ by $1$ .

Opposite $= - 1$

Find the value of sine.

Tap for more steps...

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

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

Substitute in the known values.

$\sin (x) = \frac{- 1}{\sqrt{10}}$

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

Tap for more steps...

Move the negative in front of the fraction.

$\sin (x) = - \frac{1}{\sqrt{10}}$

Multiply $\frac{1}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\sin (x) = - (\frac{1}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}})$

Combine and simplify the denominator.

Tap for more steps...

Multiply $\frac{1}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\sin (x) = - \frac{\sqrt{10}}{\sqrt{10} \sqrt{10}}$

Raise $\sqrt{10}$ to the power of $1$ .

$\sin (x) = - \frac{\sqrt{10}}{\sqrt{10} \sqrt{10}}$

Raise $\sqrt{10}$ to the power of $1$ .

$\sin (x) = - \frac{\sqrt{10}}{\sqrt{10} \sqrt{10}}$

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

$\sin (x) = - \frac{\sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Add $1$ and $1$ .

$\sin (x) = - \frac{\sqrt{10}}{{\sqrt{10}}^{2}}$

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Tap for more steps...

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\sin (x) = - \frac{\sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

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

$\sin (x) = - \frac{\sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

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

$\sin (x) = - \frac{\sqrt{10}}{10^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Tap for more steps...

Cancel the common factor.

$\sin (x) = - \frac{\sqrt{10}}{10^{\frac{2}{2}}}$

Rewrite the expression.

$\sin (x) = - \frac{\sqrt{10}}{10}$

Evaluate the exponent.

$\sin (x) = - \frac{\sqrt{10}}{10}$

Simplify the value of cosine.

Tap for more steps...

Multiply $\frac{3}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\cos (x) = - (\frac{3}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}})$

Combine and simplify the denominator.

Tap for more steps...

Multiply $\frac{3}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\cos (x) = - \frac{3 \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Raise $\sqrt{10}$ to the power of $1$ .

$\cos (x) = - \frac{3 \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Raise $\sqrt{10}$ to the power of $1$ .

$\cos (x) = - \frac{3 \sqrt{10}}{\sqrt{10} \sqrt{10}}$

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

$\cos (x) = - \frac{3 \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Add $1$ and $1$ .

$\cos (x) = - \frac{3 \sqrt{10}}{{\sqrt{10}}^{2}}$

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Tap for more steps...

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\cos (x) = - \frac{3 \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

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

$\cos (x) = - \frac{3 \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

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

$\cos (x) = - \frac{3 \sqrt{10}}{10^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Tap for more steps...

Cancel the common factor.

$\cos (x) = - \frac{3 \sqrt{10}}{10^{\frac{2}{2}}}$

Rewrite the expression.

$\cos (x) = - \frac{3 \sqrt{10}}{10}$

Evaluate the exponent.

$\cos (x) = - \frac{3 \sqrt{10}}{10}$

Find the value of tangent.

Tap for more steps...

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

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

Substitute in the known values.

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

Dividing two negative values results in a positive value.

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

Find the value of cotangent.

Tap for more steps...

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

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

Substitute in the known values.

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

Divide $- 3$ by $- 1$ .

$\cot (x) = 3$

Find the value of secant.

Tap for more steps...

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

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

Substitute in the known values.

$\sec (x) = \frac{\sqrt{10}}{- 3}$

Move the negative in front of the fraction.

$\sec (x) = - \frac{\sqrt{10}}{3}$

Find the value of cosecant.

Tap for more steps...

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

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

Substitute in the known values.

$\csc (x) = \frac{\sqrt{10}}{- 1}$

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

Tap for more steps...

Move the negative one from the denominator of $\frac{\sqrt{10}}{- 1}$ .

$\csc (x) = - 1 \cdot \sqrt{10}$

Rewrite $- 1 \cdot \sqrt{10}$ as $- \sqrt{10}$ .

$\csc (x) = - \sqrt{10}$

This is the solution to each trig value.

$\sin (x) = - \frac{\sqrt{10}}{10}$

$\cos (x) = - \frac{3 \sqrt{10}}{10}$

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

$\cot (x) = 3$

$\sec (x) = - \frac{\sqrt{10}}{3}$

$\csc (x) = - \sqrt{10}$