Trigonometry Examples

$\sin (θ) \cos (θ) - \frac{1}{2} \cos (θ) = 0$

Step 1

Divide each term in the equation by $\cos (θ)$ .

$\frac{\sin (θ) \cos (θ)}{\cos (θ)} + \frac{- \frac{1}{2} \cdot \cos (θ)}{\cos (θ)} = \frac{0}{\cos (θ)}$

Step 2

Cancel the common factor of $\cos (θ)$ .

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Step 2.1

Cancel the common factor.

$\frac{\sin (θ) \cos (θ)}{\cos (θ)} + \frac{- \frac{1}{2} \cdot \cos (θ)}{\cos (θ)} = \frac{0}{\cos (θ)}$

Step 2.2

Divide $\sin (θ)$ by $1$ .

$\sin (θ) + \frac{- \frac{1}{2} \cdot \cos (θ)}{\cos (θ)} = \frac{0}{\cos (θ)}$

Step 3

Cancel the common factor of $\cos (θ)$ .

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Step 3.1

Cancel the common factor.

$\sin (θ) + \frac{- \frac{1}{2} \cdot \cos (θ)}{\cos (θ)} = \frac{0}{\cos (θ)}$

Step 3.2

Divide $- \frac{1}{2}$ by $1$ .

$\sin (θ) - \frac{1}{2} = \frac{0}{\cos (θ)}$

Step 4

Separate fractions.

$\sin (θ) - \frac{1}{2} = \frac{0}{1} \cdot \frac{1}{\cos (θ)}$

Step 5

Convert from $\frac{1}{\cos (θ)}$ to $\sec (θ)$ .

$\sin (θ) - \frac{1}{2} = \frac{0}{1} \cdot \sec (θ)$

Step 6

Divide $0$ by $1$ .

$\sin (θ) - \frac{1}{2} = 0 \sec (θ)$

Step 7

Multiply $0$ by $\sec (θ)$ .

$\sin (θ) - \frac{1}{2} = 0$

Step 8

Add $\frac{1}{2}$ to both sides of the equation.

$\sin (θ) = \frac{1}{2}$

Step 9

Take the inverse sine of both sides of the equation to extract $θ$ from inside the sine.

$θ = \arcsin (\frac{1}{2})$

Step 10

Simplify the right side.

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Step 10.1

The exact value of $\arcsin (\frac{1}{2})$ is $30$ .

$θ = 30$

Step 11

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the second quadrant.

$θ = 180 - 30$

Step 12

Subtract $30$ from $180$ .

$θ = 150$

Step 13

Find the period of $\sin (θ)$ .

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Step 13.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 13.2

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

$\frac{360}{| 1 |}$

Step 13.3

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

$\frac{360}{1}$

Step 13.4

Divide $360$ by $1$ .

$360$

Step 14

The period of the $\sin (θ)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$θ = 30 + 360 n, 150 + 360 n$ , for any integer $n$