Trigonometry Examples

$\cos (60) = 1 - \sin^{2} (30)$

Step 1

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

$\frac{1}{2} = 1 - \sin^{2} (30)$

Step 2

Simplify $1 - \sin^{2} (30)$ .

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Step 2.1

Apply pythagorean identity.

$\frac{1}{2} = \cos^{2} (30)$

Step 2.2

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

$\frac{1}{2} = {(\frac{\sqrt{3}}{2})}^{2}$

Step 2.3

Apply the product rule to $\frac{\sqrt{3}}{2}$ .

$\frac{1}{2} = \frac{{\sqrt{3}}^{2}}{2^{2}}$

Step 2.4

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{1}{2} = \frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}}$

Step 2.4.2

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

$\frac{1}{2} = \frac{3^{\frac{1}{2} \cdot 2}}{2^{2}}$

Step 2.4.3

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

$\frac{1}{2} = \frac{3^{\frac{2}{2}}}{2^{2}}$

Step 2.4.4

Cancel the common factor of $2$ .

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Step 2.4.4.1

Cancel the common factor.

$\frac{1}{2} = \frac{3^{\frac{2}{2}}}{2^{2}}$

Step 2.4.4.2

Rewrite the expression.

$\frac{1}{2} = \frac{3^{1}}{2^{2}}$

Step 2.4.5

Evaluate the exponent.

$\frac{1}{2} = \frac{3}{2^{2}}$

Step 2.5

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

$\frac{1}{2} = \frac{3}{4}$

Step 3