Trigonometry Examples

$\sin^{2} (45 °) + \cos^{2} (30 °)$

Step 1

Simplify each term.

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Step 1.1

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

${(\frac{\sqrt{2}}{2})}^{2} + \cos^{2} (30 °)$

Step 1.2

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

$\frac{{\sqrt{2}}^{2}}{2^{2}} + \cos^{2} (30 °)$

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

$\frac{2^{\frac{1}{2} \cdot 2}}{2^{2}} + \cos^{2} (30 °)$

Step 1.3.3

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

$\frac{2^{\frac{2}{2}}}{2^{2}} + \cos^{2} (30 °)$

Step 1.3.4

Cancel the common factor of $2$ .

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Step 1.3.4.1

Cancel the common factor.

$\frac{2^{\frac{2}{2}}}{2^{2}} + \cos^{2} (30 °)$

Step 1.3.4.2

Rewrite the expression.

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

Step 1.3.5

Evaluate the exponent.

$\frac{2}{2^{2}} + \cos^{2} (30 °)$

Step 1.4

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

$\frac{2}{4} + \cos^{2} (30 °)$

Step 1.5

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

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Step 1.5.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{4} + \cos^{2} (30 °)$

Step 1.5.2

Cancel the common factors.

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Step 1.5.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 1}{2 \cdot 2} + \cos^{2} (30 °)$

Step 1.5.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 2} + \cos^{2} (30 °)$

Step 1.5.2.3

Rewrite the expression.

$\frac{1}{2} + \cos^{2} (30 °)$

Step 1.6

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

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

Step 1.7

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

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

Step 1.8

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

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Step 1.8.1

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 1.8.2

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 1.8.3

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

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

Step 1.8.4

Cancel the common factor of $2$ .

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Step 1.8.4.1

Cancel the common factor.

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

Step 1.8.4.2

Rewrite the expression.

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

Step 1.8.5

Evaluate the exponent.

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

Step 1.9

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

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

Step 2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} \cdot \frac{2}{2} + \frac{3}{4}$

Step 3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

$\frac{2}{2 \cdot 2} + \frac{3}{4}$

Step 3.2

Multiply $2$ by $2$ .

$\frac{2}{4} + \frac{3}{4}$

Step 4

Combine the numerators over the common denominator.

$\frac{2 + 3}{4}$

Step 5

Add $2$ and $3$ .

$\frac{5}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{4}$

Decimal Form:

$1.25$

Mixed Number Form:

$1 \frac{1}{4}$