Trigonometry Examples

$\sin^{2} (225)$

Step 1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the third quadrant.

${(- \sin (45))}^{2}$

Step 2

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

${(- \frac{\sqrt{2}}{2})}^{2}$

Step 3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1

Apply the product rule to $- \frac{\sqrt{2}}{2}$ .

${(- 1)}^{2} {(\frac{\sqrt{2}}{2})}^{2}$

Step 3.2

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

${(- 1)}^{2} \frac{{\sqrt{2}}^{2}}{2^{2}}$

Step 4

Simplify the expression.

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Step 4.1

Raise $- 1$ to the power of $2$ .

$1 \frac{{\sqrt{2}}^{2}}{2^{2}}$

Step 4.2

Multiply $\frac{{\sqrt{2}}^{2}}{2^{2}}$ by $1$ .

$\frac{{\sqrt{2}}^{2}}{2^{2}}$

Step 5

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

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Step 5.1

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

$\frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}}$

Step 5.2

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

$\frac{2^{\frac{1}{2} \cdot 2}}{2^{2}}$

Step 5.3

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

$\frac{2^{\frac{2}{2}}}{2^{2}}$

Step 5.4

Cancel the common factor of $2$ .

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Step 5.4.1

Cancel the common factor.

$\frac{2^{\frac{2}{2}}}{2^{2}}$

Step 5.4.2

Rewrite the expression.

$\frac{2^{1}}{2^{2}}$

Step 5.5

Evaluate the exponent.

$\frac{2}{2^{2}}$

Step 6

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

$\frac{2}{4}$

Step 7

Cancel the common factor of $2$ and $4$ .

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Step 7.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{4}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 1}{2 \cdot 2}$

Step 7.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 2}$

Step 7.2.3

Rewrite the expression.

$\frac{1}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$