Trigonometry Examples

$\sin (x) \cos (x) \tan (x) = \sin^{2} (0)$

Step 1

Subtract $\sin^{2} (0)$ from both sides of the equation.

$\sin (x) \cos (x) \tan (x) - \sin^{2} (0) = 0$

Step 2

Simplify $\sin (x) \cos (x) \tan (x) - \sin^{2} (0)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

$\sin (x) \cos (x) \frac{\sin (x)}{\cos (x)} - \sin^{2} (0) = 0$

Step 2.1.2

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

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Step 2.1.2.1

Factor $\cos (x)$ out of $\sin (x) \cos (x)$ .

$\cos (x) \sin (x) \frac{\sin (x)}{\cos (x)} - \sin^{2} (0) = 0$

Step 2.1.2.2

Cancel the common factor.

$\cos (x) \sin (x) \frac{\sin (x)}{\cos (x)} - \sin^{2} (0) = 0$

Step 2.1.2.3

Rewrite the expression.

$\sin (x) \sin (x) - \sin^{2} (0) = 0$

Step 2.1.3

Raise $\sin (x)$ to the power of $1$ .

$\sin^{1} (x) \sin (x) - \sin^{2} (0) = 0$

Step 2.1.4

Raise $\sin (x)$ to the power of $1$ .

$\sin^{1} (x) \sin^{1} (x) - \sin^{2} (0) = 0$

Step 2.1.5

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

${\sin (x)}^{1 + 1} - \sin^{2} (0) = 0$

Step 2.1.6

Add $1$ and $1$ .

$\sin^{2} (x) - \sin^{2} (0) = 0$

Step 2.1.7

The exact value of $\sin (0)$ is $0$ .

$\sin^{2} (x) - 0^{2} = 0$

Step 2.1.8

Raising $0$ to any positive power yields $0$ .

$\sin^{2} (x) - 0 = 0$

Step 2.1.9

Multiply $- 1$ by $0$ .

$\sin^{2} (x) + 0 = 0$

Step 2.2

Add $\sin^{2} (x)$ and $0$ .

$\sin^{2} (x) = 0$

Step 3

Solve for $x$ .

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Step 3.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\sin (x) = \pm \sqrt{0}$

Step 3.2

Simplify $\pm \sqrt{0}$ .

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Step 3.2.1

Rewrite $0$ as $0^{2}$ .

$\sin (x) = \pm \sqrt{0^{2}}$

Step 3.2.2

Pull terms out from under the radical, assuming positive real numbers.

$\sin (x) = \pm 0$

Step 3.2.3

Plus or minus $0$ is $0$ .

$\sin (x) = 0$

Step 3.3

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

$x = \arcsin (0)$

Step 3.4

Simplify the right side.

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Step 3.4.1

The exact value of $\arcsin (0)$ is $0$ .

$x = 0$

Step 3.5

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.

$x = 180 - 0$

Step 3.6

Subtract $0$ from $180$ .

$x = 180$

Step 3.7

Find the period of $\sin (x)$ .

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Step 3.7.1

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

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

Step 3.7.2

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

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

Step 3.7.3

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

$\frac{360}{1}$

Step 3.7.4

Divide $360$ by $1$ .

$360$

Step 3.8

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

$x = 360 n, 180 + 360 n$ , for any integer $n$

Step 4

Consolidate the answers.

$x = 180 n$ , for any integer $n$