Trigonometry Examples

$6 \sqrt{2} \cos (x + 1) = 7$

Step 1

Divide each term in $6 \sqrt{2} \cos (x + 1) = 7$ by $6 \sqrt{2}$ and simplify.

Tap for more steps...

Step 1.1

Divide each term in $6 \sqrt{2} \cos (x + 1) = 7$ by $6 \sqrt{2}$ .

$\frac{6 \sqrt{2} \cos (x + 1)}{6 \sqrt{2}} = \frac{7}{6 \sqrt{2}}$

Step 1.2

Simplify the left side.

Tap for more steps...

Step 1.2.1

Cancel the common factor of $6$ .

Tap for more steps...

Step 1.2.1.1

Cancel the common factor.

$\frac{6 \sqrt{2} \cos (x + 1)}{6 \sqrt{2}} = \frac{7}{6 \sqrt{2}}$

Step 1.2.1.2

Rewrite the expression.

$\frac{\sqrt{2} \cos (x + 1)}{\sqrt{2}} = \frac{7}{6 \sqrt{2}}$

Step 1.2.2

Cancel the common factor of $\sqrt{2}$ .

Tap for more steps...

Step 1.2.2.1

Cancel the common factor.

$\frac{\sqrt{2} \cos (x + 1)}{\sqrt{2}} = \frac{7}{6 \sqrt{2}}$

Step 1.2.2.2

Divide $\cos (x + 1)$ by $1$ .

$\cos (x + 1) = \frac{7}{6 \sqrt{2}}$

Step 1.3

Simplify the right side.

Tap for more steps...

Step 1.3.1

Multiply $\frac{7}{6 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\cos (x + 1) = \frac{7}{6 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 1.3.2

Combine and simplify the denominator.

Tap for more steps...

Step 1.3.2.1

Multiply $\frac{7}{6 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\cos (x + 1) = \frac{7 \sqrt{2}}{6 \sqrt{2} \sqrt{2}}$

Step 1.3.2.2

Move $\sqrt{2}$ .

$\cos (x + 1) = \frac{7 \sqrt{2}}{6 (\sqrt{2} \sqrt{2})}$

Step 1.3.2.3

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

$\cos (x + 1) = \frac{7 \sqrt{2}}{6 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 1.3.2.4

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

$\cos (x + 1) = \frac{7 \sqrt{2}}{6 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 1.3.2.5

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

$\cos (x + 1) = \frac{7 \sqrt{2}}{6 {\sqrt{2}}^{1 + 1}}$

Step 1.3.2.6

Add $1$ and $1$ .

$\cos (x + 1) = \frac{7 \sqrt{2}}{6 {\sqrt{2}}^{2}}$

Step 1.3.2.7

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

Tap for more steps...

Step 1.3.2.7.1

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

$\cos (x + 1) = \frac{7 \sqrt{2}}{6 {(2^{\frac{1}{2}})}^{2}}$

Step 1.3.2.7.2

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

$\cos (x + 1) = \frac{7 \sqrt{2}}{6 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 1.3.2.7.3

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

$\cos (x + 1) = \frac{7 \sqrt{2}}{6 \cdot 2^{\frac{2}{2}}}$

Step 1.3.2.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.3.2.7.4.1

Cancel the common factor.

$\cos (x + 1) = \frac{7 \sqrt{2}}{6 \cdot 2^{\frac{2}{2}}}$

Step 1.3.2.7.4.2

Rewrite the expression.

$\cos (x + 1) = \frac{7 \sqrt{2}}{6 \cdot 2^{1}}$

Step 1.3.2.7.5

Evaluate the exponent.

$\cos (x + 1) = \frac{7 \sqrt{2}}{6 \cdot 2}$

Step 1.3.3

Multiply $6$ by $2$ .

$\cos (x + 1) = \frac{7 \sqrt{2}}{12}$

Step 2

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x + 1 = \arccos (\frac{7 \sqrt{2}}{12})$

Step 3

Simplify the right side.

Tap for more steps...

Step 3.1

Evaluate $\arccos (\frac{7 \sqrt{2}}{12})$ .

$x + 1 = 0.60066859$

Step 4

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 4.1

Subtract $1$ from both sides of the equation.

$x = 0.60066859 - 1$

Step 4.2

Subtract $1$ from $0.60066859$ .

$x = - 0.3993314$

Step 5

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x + 1 = 2 (3.14159265) - 0.60066859$

Step 6

Solve for $x$ .

Tap for more steps...

Step 6.1

Simplify $2 (3.14159265) - 0.60066859$ .

Tap for more steps...

Step 6.1.1

Multiply $2$ by $3.14159265$ .

$x + 1 = 6.2831853 - 0.60066859$

Step 6.1.2

Subtract $0.60066859$ from $6.2831853$ .

$x + 1 = 5.6825167$

Step 6.2

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 6.2.1

Subtract $1$ from both sides of the equation.

$x = 5.6825167 - 1$

Step 6.2.2

Subtract $1$ from $5.6825167$ .

$x = 4.6825167$

Step 7

Find the period of $\cos (x + 1)$ .

Tap for more steps...

Step 7.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 7.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 7.4

Divide $2 π$ by $1$ .

$2 π$

Step 8

Add $2 π$ to every negative angle to get positive angles.

Tap for more steps...

Step 8.1

Add $2 π$ to $- 0.3993314$ to find the positive angle.

$- 0.3993314 + 2 π$

Step 8.2

Subtract $0.3993314$ from $2 π$ .

$5.8838539$

Step 8.3

List the new angles.

$x = 5.8838539$

Step 9

The period of the $\cos (x + 1)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 4.6825167 + 2 π n, 5.8838539 + 2 π n$ , for any integer $n$