Trigonometry Examples

$\cos (- x) = - \frac{\sqrt{2}}{5}$

Step 1

Use the opposite angle identity that states that $c o s (- x) = c o s (x)$ .

$\cos (x) = - \frac{\sqrt{2}}{5}$

Step 2

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.

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

Step 3

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 4

Replace the known values in the equation.

$Opposite = \sqrt{{(5)}^{2} - {(- \sqrt{2})}^{2}}$

Step 5

Simplify inside the radical.

Tap for more steps...

Step 5.1

Raise $5$ to the power of $2$ .

Opposite $= \sqrt{25 - {(- \sqrt{2})}^{2}}$

Step 5.2

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

Opposite $= \sqrt{25 - ({(- 1)}^{2} {\sqrt{2}}^{2})}$

Step 5.3

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

Tap for more steps...

Step 5.3.1

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

Opposite $= \sqrt{25 + {(- 1)}^{2} \cdot (- 1 {\sqrt{2}}^{2})}$

Step 5.3.2

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

Tap for more steps...

Step 5.3.2.1

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

Opposite $= \sqrt{25 + {(- 1)}^{2} \cdot ((- 1) {\sqrt{2}}^{2})}$

Step 5.3.2.2

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

Opposite $= \sqrt{25 + {(- 1)}^{2 + 1} {\sqrt{2}}^{2}}$

Step 5.3.3

Add $2$ and $1$ .

Opposite $= \sqrt{25 + {(- 1)}^{3} {\sqrt{2}}^{2}}$

Step 5.4

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

Opposite $= \sqrt{25 - {\sqrt{2}}^{2}}$

Step 5.5

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

Tap for more steps...

Step 5.5.1

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

Opposite $= \sqrt{25 - {(2^{\frac{1}{2}})}^{2}}$

Step 5.5.2

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

Opposite $= \sqrt{25 - 2^{\frac{1}{2} \cdot 2}}$

Step 5.5.3

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

Opposite $= \sqrt{25 - 2^{\frac{2}{2}}}$

Step 5.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.5.4.1

Cancel the common factor.

Opposite $= \sqrt{25 - 2^{\frac{2}{2}}}$

Step 5.5.4.2

Rewrite the expression.

Opposite $= \sqrt{25 - 2}$

Step 5.5.5

Evaluate the exponent.

Opposite $= \sqrt{25 - 1 \cdot 2}$

Step 5.6

Multiply $- 1$ by $2$ .

Opposite $= \sqrt{25 - 2}$

Step 5.7

Subtract $2$ from $25$ .

Opposite $= \sqrt{23}$

Step 6

Find the value of sine.

Tap for more steps...

Step 6.1

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

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

Step 6.2

Substitute in the known values.

$\sin (x) = \frac{\sqrt{23}}{5}$

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.

$\tan (x) = \frac{\sqrt{23}}{- \sqrt{2}}$

Step 7.3

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

Tap for more steps...

Step 7.3.1

Move the negative in front of the fraction.

$\tan (x) = - \frac{\sqrt{23}}{\sqrt{2}}$

Step 7.3.2

Multiply $\frac{\sqrt{23}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\tan (x) = - (\frac{\sqrt{23}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 7.3.3

Combine and simplify the denominator.

Tap for more steps...

Step 7.3.3.1

Multiply $\frac{\sqrt{23}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\tan (x) = - \frac{\sqrt{23} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 7.3.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$\tan (x) = - \frac{\sqrt{23} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 7.3.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$\tan (x) = - \frac{\sqrt{23} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 7.3.3.4

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

$\tan (x) = - \frac{\sqrt{23} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 7.3.3.5

Add $1$ and $1$ .

$\tan (x) = - \frac{\sqrt{23} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 7.3.3.6

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

Tap for more steps...

Step 7.3.3.6.1

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

$\tan (x) = - \frac{\sqrt{23} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 7.3.3.6.2

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

$\tan (x) = - \frac{\sqrt{23} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 7.3.3.6.3

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

$\tan (x) = - \frac{\sqrt{23} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 7.3.3.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.3.3.6.4.1

Cancel the common factor.

$\tan (x) = - \frac{\sqrt{23} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 7.3.3.6.4.2

Rewrite the expression.

$\tan (x) = - \frac{\sqrt{23} \sqrt{2}}{2}$

Step 7.3.3.6.5

Evaluate the exponent.

$\tan (x) = - \frac{\sqrt{23} \sqrt{2}}{2}$

Step 7.3.4

Simplify the numerator.

Tap for more steps...

Step 7.3.4.1

Combine using the product rule for radicals.

$\tan (x) = - \frac{\sqrt{23 \cdot 2}}{2}$

Step 7.3.4.2

Multiply $23$ by $2$ .

$\tan (x) = - \frac{\sqrt{46}}{2}$

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.

$\cot (x) = \frac{- \sqrt{2}}{\sqrt{23}}$

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.

$\cot (x) = - \frac{\sqrt{2}}{\sqrt{23}}$

Step 8.3.2

Multiply $\frac{\sqrt{2}}{\sqrt{23}}$ by $\frac{\sqrt{23}}{\sqrt{23}}$ .

$\cot (x) = - (\frac{\sqrt{2}}{\sqrt{23}} \cdot \frac{\sqrt{23}}{\sqrt{23}})$

Step 8.3.3

Combine and simplify the denominator.

Tap for more steps...

Step 8.3.3.1

Multiply $\frac{\sqrt{2}}{\sqrt{23}}$ by $\frac{\sqrt{23}}{\sqrt{23}}$ .

$\cot (x) = - \frac{\sqrt{2} \sqrt{23}}{\sqrt{23} \sqrt{23}}$

Step 8.3.3.2

Raise $\sqrt{23}$ to the power of $1$ .

$\cot (x) = - \frac{\sqrt{2} \sqrt{23}}{\sqrt{23} \sqrt{23}}$

Step 8.3.3.3

Raise $\sqrt{23}$ to the power of $1$ .

$\cot (x) = - \frac{\sqrt{2} \sqrt{23}}{\sqrt{23} \sqrt{23}}$

Step 8.3.3.4

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

$\cot (x) = - \frac{\sqrt{2} \sqrt{23}}{{\sqrt{23}}^{1 + 1}}$

Step 8.3.3.5

Add $1$ and $1$ .

$\cot (x) = - \frac{\sqrt{2} \sqrt{23}}{{\sqrt{23}}^{2}}$

Step 8.3.3.6

Rewrite ${\sqrt{23}}^{2}$ as $23$ .

Tap for more steps...

Step 8.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{23}$ as $23^{\frac{1}{2}}$ .

$\cot (x) = - \frac{\sqrt{2} \sqrt{23}}{{(23^{\frac{1}{2}})}^{2}}$

Step 8.3.3.6.2

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

$\cot (x) = - \frac{\sqrt{2} \sqrt{23}}{23^{\frac{1}{2} \cdot 2}}$

Step 8.3.3.6.3

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

$\cot (x) = - \frac{\sqrt{2} \sqrt{23}}{23^{\frac{2}{2}}}$

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.

$\cot (x) = - \frac{\sqrt{2} \sqrt{23}}{23^{\frac{2}{2}}}$

Step 8.3.3.6.4.2

Rewrite the expression.

$\cot (x) = - \frac{\sqrt{2} \sqrt{23}}{23}$

Step 8.3.3.6.5

Evaluate the exponent.

$\cot (x) = - \frac{\sqrt{2} \sqrt{23}}{23}$

Step 8.3.4

Simplify the numerator.

Tap for more steps...

Step 8.3.4.1

Combine using the product rule for radicals.

$\cot (x) = - \frac{\sqrt{2 \cdot 23}}{23}$

Step 8.3.4.2

Multiply $2$ by $23$ .

$\cot (x) = - \frac{\sqrt{46}}{23}$

Step 9

Find the value of secant.

Tap for more steps...

Step 9.1

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

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

Step 9.2

Substitute in the known values.

$\sec (x) = \frac{5}{- \sqrt{2}}$

Step 9.3

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

Tap for more steps...

Step 9.3.1

Move the negative in front of the fraction.

$\sec (x) = - \frac{5}{\sqrt{2}}$

Step 9.3.2

Multiply $\frac{5}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sec (x) = - (\frac{5}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 9.3.3

Combine and simplify the denominator.

Tap for more steps...

Step 9.3.3.1

Multiply $\frac{5}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sec (x) = - \frac{5 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 9.3.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$\sec (x) = - \frac{5 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 9.3.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$\sec (x) = - \frac{5 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 9.3.3.4

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

$\sec (x) = - \frac{5 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 9.3.3.5

Add $1$ and $1$ .

$\sec (x) = - \frac{5 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 9.3.3.6

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

Tap for more steps...

Step 9.3.3.6.1

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

$\sec (x) = - \frac{5 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 9.3.3.6.2

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

$\sec (x) = - \frac{5 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 9.3.3.6.3

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

$\sec (x) = - \frac{5 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 9.3.3.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.3.3.6.4.1

Cancel the common factor.

$\sec (x) = - \frac{5 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 9.3.3.6.4.2

Rewrite the expression.

$\sec (x) = - \frac{5 \sqrt{2}}{2}$

Step 9.3.3.6.5

Evaluate the exponent.

$\sec (x) = - \frac{5 \sqrt{2}}{2}$

Step 10

Find the value of cosecant.

Tap for more steps...

Step 10.1

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

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

Step 10.2

Substitute in the known values.

$\csc (x) = \frac{5}{\sqrt{23}}$

Step 10.3

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

Tap for more steps...

Step 10.3.1

Multiply $\frac{5}{\sqrt{23}}$ by $\frac{\sqrt{23}}{\sqrt{23}}$ .

$\csc (x) = \frac{5}{\sqrt{23}} \cdot \frac{\sqrt{23}}{\sqrt{23}}$

Step 10.3.2

Combine and simplify the denominator.

Tap for more steps...

Step 10.3.2.1

Multiply $\frac{5}{\sqrt{23}}$ by $\frac{\sqrt{23}}{\sqrt{23}}$ .

$\csc (x) = \frac{5 \sqrt{23}}{\sqrt{23} \sqrt{23}}$

Step 10.3.2.2

Raise $\sqrt{23}$ to the power of $1$ .

$\csc (x) = \frac{5 \sqrt{23}}{\sqrt{23} \sqrt{23}}$

Step 10.3.2.3

Raise $\sqrt{23}$ to the power of $1$ .

$\csc (x) = \frac{5 \sqrt{23}}{\sqrt{23} \sqrt{23}}$

Step 10.3.2.4

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

$\csc (x) = \frac{5 \sqrt{23}}{{\sqrt{23}}^{1 + 1}}$

Step 10.3.2.5

Add $1$ and $1$ .

$\csc (x) = \frac{5 \sqrt{23}}{{\sqrt{23}}^{2}}$

Step 10.3.2.6

Rewrite ${\sqrt{23}}^{2}$ as $23$ .

Tap for more steps...

Step 10.3.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{23}$ as $23^{\frac{1}{2}}$ .

$\csc (x) = \frac{5 \sqrt{23}}{{(23^{\frac{1}{2}})}^{2}}$

Step 10.3.2.6.2

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

$\csc (x) = \frac{5 \sqrt{23}}{23^{\frac{1}{2} \cdot 2}}$

Step 10.3.2.6.3

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

$\csc (x) = \frac{5 \sqrt{23}}{23^{\frac{2}{2}}}$

Step 10.3.2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 10.3.2.6.4.1

Cancel the common factor.

$\csc (x) = \frac{5 \sqrt{23}}{23^{\frac{2}{2}}}$

Step 10.3.2.6.4.2

Rewrite the expression.

$\csc (x) = \frac{5 \sqrt{23}}{23}$

Step 10.3.2.6.5

Evaluate the exponent.

$\csc (x) = \frac{5 \sqrt{23}}{23}$

Step 11

This is the solution to each trig value.

$\sin (x) = \frac{\sqrt{23}}{5}$

$\cos (x) = - \frac{\sqrt{2}}{5}$

$\tan (x) = - \frac{\sqrt{46}}{2}$

$\cot (x) = - \frac{\sqrt{46}}{23}$

$\sec (x) = - \frac{5 \sqrt{2}}{2}$

$\csc (x) = \frac{5 \sqrt{23}}{23}$