Trigonometry Examples

$x^{2} = 7^{2} + 9^{2} - 2 \cdot 7 \cdot 9 \cos (30)$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $7^{2}$ from both sides of the equation.

$x^{2} - 7^{2} = 9^{2} - 2 \cdot 7 \cdot 9 \cos (30)$

Step 1.2

Subtract $9^{2}$ from both sides of the equation.

$x^{2} - 7^{2} - 9^{2} = - 2 \cdot 7 \cdot 9 \cos (30)$

Step 1.3

Add $2 \cdot 7 \cdot 9 \cos (30)$ to both sides of the equation.

$x^{2} - 7^{2} - 9^{2} + 2 \cdot 7 \cdot 9 \cos (30) = 0$

Step 2

Simplify $x^{2} - 7^{2} - 9^{2} + 2 \cdot 7 \cdot 9 \cos (30)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Raise $7$ to the power of $2$ .

$x^{2} - 1 \cdot 49 - 9^{2} + 2 \cdot 7 \cdot 9 \cos (30) = 0$

Step 2.1.2

Multiply $- 1$ by $49$ .

$x^{2} - 49 - 9^{2} + 2 \cdot 7 \cdot 9 \cos (30) = 0$

Step 2.1.3

Raise $9$ to the power of $2$ .

$x^{2} - 49 - 1 \cdot 81 + 2 \cdot 7 \cdot 9 \cos (30) = 0$

Step 2.1.4

Multiply $- 1$ by $81$ .

$x^{2} - 49 - 81 + 2 \cdot 7 \cdot 9 \cos (30) = 0$

Step 2.1.5

Multiply $2 \cdot 7 \cdot 9$ .

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Step 2.1.5.1

Multiply $2$ by $7$ .

$x^{2} - 49 - 81 + 14 \cdot 9 \cos (30) = 0$

Step 2.1.5.2

Multiply $14$ by $9$ .

$x^{2} - 49 - 81 + 126 \cos (30) = 0$

Step 2.1.6

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$x^{2} - 49 - 81 + 126 \frac{\sqrt{3}}{2} = 0$

Step 2.1.7

Cancel the common factor of $2$ .

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Step 2.1.7.1

Factor $2$ out of $126$ .

$x^{2} - 49 - 81 + 2 (63) \frac{\sqrt{3}}{2} = 0$

Step 2.1.7.2

Cancel the common factor.

$x^{2} - 49 - 81 + 2 \cdot 63 \frac{\sqrt{3}}{2} = 0$

Step 2.1.7.3

Rewrite the expression.

$x^{2} - 49 - 81 + 63 \sqrt{3} = 0$

Step 2.2

Subtract $81$ from $- 49$ .

$x^{2} - 130 + 63 \sqrt{3} = 0$

Step 3

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.