Trigonometry Examples

{sin}^{2} (45) + {cos}^{2} (60)

$\sin^{2} (45) + \cos^{2} (60)$

Step 1

Simplify each term.

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Step 1.1

The exact value of

sin (45)

$\sin (45)$ is

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

{(\frac{\sqrt{2}}{2})}^{2} + {cos}^{2} (60)

${(\frac{\sqrt{2}}{2})}^{2} + \cos^{2} (60)$

Step 1.2

Apply the product rule to

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

\frac{{\sqrt{2}}^{2}}{2^{2}} + {cos}^{2} (60)

$\frac{{\sqrt{2}}^{2}}{2^{2}} + \cos^{2} (60)$

Step 1.3

Rewrite

{\sqrt{2}}^{2}

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

2

$2$ .

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Step 1.3.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{2}

$\sqrt{2}$ as

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

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

\frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}} + {cos}^{2} (60)

$\frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}} + \cos^{2} (60)$

Step 1.3.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\frac{2^{\frac{1}{2} \cdot 2}}{2^{2}} + {cos}^{2} (60)

$\frac{2^{\frac{1}{2} \cdot 2}}{2^{2}} + \cos^{2} (60)$

Step 1.3.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\frac{2^{\frac{2}{2}}}{2^{2}} + {cos}^{2} (60)

$\frac{2^{\frac{2}{2}}}{2^{2}} + \cos^{2} (60)$

Step 1.3.4

Cancel the common factor of

2

$2$ .

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Step 1.3.4.1

Cancel the common factor.

$\frac{2^{\frac{2}{2}}}{2^{2}} + \cos^{2} (60)$

Step 1.3.4.2

Rewrite the expression.

$\frac{2^{1}}{2^{2}} + \cos^{2} (60)$

Step 1.3.5

Evaluate the exponent.

$\frac{2}{2^{2}} + \cos^{2} (60)$

Step 1.4

Raise

$2$ to the power of

$2$ .

$\frac{2}{4} + \cos^{2} (60)$

Step 1.5

Cancel the common factor of

$2$ and

$4$ .

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Step 1.5.1

Factor

$2$ out of

$2$ .

$\frac{2 (1)}{4} + \cos^{2} (60)$

Step 1.5.2

Cancel the common factors.

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Step 1.5.2.1

Factor

$2$ out of

$4$ .

$\frac{2 \cdot 1}{2 \cdot 2} + \cos^{2} (60)$

Step 1.5.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 2} + \cos^{2} (60)$

Step 1.5.2.3

Rewrite the expression.

$\frac{1}{2} + \cos^{2} (60)$

Step 1.6

The exact value of

$\cos (60)$ is

$\frac{1}{2}$ .

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

Step 1.7

Apply the product rule to

$\frac{1}{2}$ .

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

Step 1.8

One to any power is one.

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

Step 1.9

Raise

$2$ to the power of

$2$ .

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

Step 2

To write

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

$\frac{2}{2}$ .

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

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

$\frac{2}{2}$ .

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

Step 3.2

Multiply

$2$ by

$2$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Add

$2$ and

$1$ .

$\frac{3}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{4}$

Decimal Form:

$0.75$