Trigonometry Examples

$5 \cos (\frac{π}{4}) x + 6 = 2$

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$5 \frac{\sqrt{2}}{2} x + 6 = 2$

Step 1.1.2

Combine $5$ and $\frac{\sqrt{2}}{2}$ .

$\frac{5 \sqrt{2}}{2} x + 6 = 2$

Step 1.1.3

Combine $\frac{5 \sqrt{2}}{2}$ and $x$ .

$\frac{5 \sqrt{2} x}{2} + 6 = 2$

Step 2

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

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Step 2.1

Subtract $6$ from both sides of the equation.

$\frac{5 \sqrt{2} x}{2} = 2 - 6$

Step 2.2

Subtract $6$ from $2$ .

$\frac{5 \sqrt{2} x}{2} = - 4$

Step 3

Multiply both sides of the equation by $\frac{2}{5 \sqrt{2}}$ .

$\frac{2}{5 \sqrt{2}} \cdot \frac{5 \sqrt{2} x}{2} = \frac{2}{5 \sqrt{2}} \cdot - 4$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $\frac{2}{5 \sqrt{2}} \cdot \frac{5 \sqrt{2} x}{2}$ .

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Step 4.1.1.1

Cancel the common factor of $2$ .

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Step 4.1.1.1.1

Cancel the common factor.

$\frac{2}{5 \sqrt{2}} \cdot \frac{5 \sqrt{2} x}{2} = \frac{2}{5 \sqrt{2}} \cdot - 4$

Step 4.1.1.1.2

Rewrite the expression.

$\frac{1}{5 \sqrt{2}} (5 \sqrt{2} x) = \frac{2}{5 \sqrt{2}} \cdot - 4$

Step 4.1.1.2

Cancel the common factor of $5 \sqrt{2}$ .

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Step 4.1.1.2.1

Factor $5 \sqrt{2}$ out of $5 \sqrt{2} x$ .

$\frac{1}{5 \sqrt{2}} (5 \sqrt{2} (x)) = \frac{2}{5 \sqrt{2}} \cdot - 4$

Step 4.1.1.2.2

Cancel the common factor.

$\frac{1}{5 \sqrt{2}} (5 \sqrt{2} x) = \frac{2}{5 \sqrt{2}} \cdot - 4$

Step 4.1.1.2.3

Rewrite the expression.

$x = \frac{2}{5 \sqrt{2}} \cdot - 4$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{2}{5 \sqrt{2}} \cdot - 4$ .

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Step 4.2.1.1

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

$x = \frac{2}{5 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} \cdot - 4$

Step 4.2.1.2

Combine and simplify the denominator.

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Step 4.2.1.2.1

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

$x = \frac{2 \sqrt{2}}{5 \sqrt{2} \sqrt{2}} \cdot - 4$

Step 4.2.1.2.2

Move $\sqrt{2}$ .

$x = \frac{2 \sqrt{2}}{5 (\sqrt{2} \sqrt{2})} \cdot - 4$

Step 4.2.1.2.3

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

$x = \frac{2 \sqrt{2}}{5 ({\sqrt{2}}^{1} \sqrt{2})} \cdot - 4$

Step 4.2.1.2.4

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

$x = \frac{2 \sqrt{2}}{5 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})} \cdot - 4$

Step 4.2.1.2.5

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

$x = \frac{2 \sqrt{2}}{5 {\sqrt{2}}^{1 + 1}} \cdot - 4$

Step 4.2.1.2.6

Add $1$ and $1$ .

$x = \frac{2 \sqrt{2}}{5 {\sqrt{2}}^{2}} \cdot - 4$

Step 4.2.1.2.7

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

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Step 4.2.1.2.7.1

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

$x = \frac{2 \sqrt{2}}{5 {(2^{\frac{1}{2}})}^{2}} \cdot - 4$

Step 4.2.1.2.7.2

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

$x = \frac{2 \sqrt{2}}{5 \cdot 2^{\frac{1}{2} \cdot 2}} \cdot - 4$

Step 4.2.1.2.7.3

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

$x = \frac{2 \sqrt{2}}{5 \cdot 2^{\frac{2}{2}}} \cdot - 4$

Step 4.2.1.2.7.4

Cancel the common factor of $2$ .

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Step 4.2.1.2.7.4.1

Cancel the common factor.

$x = \frac{2 \sqrt{2}}{5 \cdot 2^{\frac{2}{2}}} \cdot - 4$

Step 4.2.1.2.7.4.2

Rewrite the expression.

$x = \frac{2 \sqrt{2}}{5 \cdot 2^{1}} \cdot - 4$

Step 4.2.1.2.7.5

Evaluate the exponent.

$x = \frac{2 \sqrt{2}}{5 \cdot 2} \cdot - 4$

Step 4.2.1.3

Multiply $5$ by $2$ .

$x = \frac{2 \sqrt{2}}{10} \cdot - 4$

Step 4.2.1.4

Cancel the common factor of $2$ .

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Step 4.2.1.4.1

Factor $2$ out of $10$ .

$x = \frac{2 \sqrt{2}}{2 (5)} \cdot - 4$

Step 4.2.1.4.2

Factor $2$ out of $- 4$ .

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

Step 4.2.1.4.3

Cancel the common factor.

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

Step 4.2.1.4.4

Rewrite the expression.

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

Step 4.2.1.5

Combine $\frac{2 \sqrt{2}}{5}$ and $- 2$ .

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

Step 4.2.1.6

Simplify the expression.

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Step 4.2.1.6.1

Multiply $- 2$ by $2$ .

$x = \frac{- 4 \sqrt{2}}{5}$

Step 4.2.1.6.2

Move the negative in front of the fraction.

$x = - \frac{4 \sqrt{2}}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{4 \sqrt{2}}{5}$

Decimal Form:

$x = - 1.13137084 \dots$