Trigonometry Examples

$\frac{\sin (60 °)}{a} = \frac{\sin (65 °)}{120}$

Step 1

Factor each term.

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Step 1.1

The exact value of $\sin (60 °)$ is $\frac{\sqrt{3}}{2}$ .

$\frac{\frac{\sqrt{3}}{2}}{a} = \frac{\sin (65 °)}{120}$

Step 1.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{3}}{2} \cdot \frac{1}{a} = \frac{\sin (65 °)}{120}$

Step 1.3

Multiply $\frac{\sqrt{3}}{2}$ by $\frac{1}{a}$ .

$\frac{\sqrt{3}}{2 a} = \frac{\sin (65 °)}{120}$

Step 1.4

Evaluate $\sin (65 °)$ .

$\frac{\sqrt{3}}{2 a} = \frac{0.90630778}{120}$

Step 1.5

Divide $0.90630778$ by $120$ .

$\frac{\sqrt{3}}{2 a} = 0.00755256$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$2 a, 1$

Step 2.2

The LCM of one and any expression is the expression.

$2 a$

Step 3

Multiply each term in $\frac{\sqrt{3}}{2 a} = 0.00755256$ by $2 a$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{\sqrt{3}}{2 a} = 0.00755256$ by $2 a$ .

$\frac{\sqrt{3}}{2 a} (2 a) = 0.00755256 (2 a)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Rewrite using the commutative property of multiplication.

$2 \frac{\sqrt{3}}{2 a} a = 0.00755256 (2 a)$

Step 3.2.2

Cancel the common factor of $2$ .

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Step 3.2.2.1

Factor $2$ out of $2 a$ .

$2 \frac{\sqrt{3}}{2 (a)} a = 0.00755256 (2 a)$

Step 3.2.2.2

Cancel the common factor.

$2 \frac{\sqrt{3}}{2 a} a = 0.00755256 (2 a)$

Step 3.2.2.3

Rewrite the expression.

$\frac{\sqrt{3}}{a} a = 0.00755256 (2 a)$

Step 3.2.3

Cancel the common factor of $a$ .

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Step 3.2.3.1

Cancel the common factor.

$\frac{\sqrt{3}}{a} a = 0.00755256 (2 a)$

Step 3.2.3.2

Rewrite the expression.

$\sqrt{3} = 0.00755256 (2 a)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $2$ by $0.00755256$ .

$\sqrt{3} = 0.01510512 a$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $0.01510512 a = \sqrt{3}$ .

$0.01510512 a = \sqrt{3}$

Step 4.2

Divide each term in $0.01510512 a = \sqrt{3}$ by $0.01510512$ and simplify.

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Step 4.2.1

Divide each term in $0.01510512 a = \sqrt{3}$ by $0.01510512$ .

$\frac{0.01510512 a}{0.01510512} = \frac{\sqrt{3}}{0.01510512}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $0.01510512$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{0.01510512 a}{0.01510512} = \frac{\sqrt{3}}{0.01510512}$

Step 4.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{\sqrt{3}}{0.01510512}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Evaluate the root.

$a = \frac{1.7320508}{0.01510512}$

Step 4.2.3.2

Divide $1.7320508$ by $0.01510512$ .

$a = 114.66639693$