Trigonometry Examples

2 \cos^{2} (157.5) - 1

$2 \cos^{2} (157.5) - 1$

Step 1

Apply the cosine double-angle identity.

\cos (2 \cdot 157.5)

$\cos (2 \cdot 157.5)$

Step 2

Multiply

2

$2$ by

157.5

$157.5$ .

\cos (315)

$\cos (315)$

Step 3

The exact value of

\cos (315)

$\cos (315)$ is

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

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

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Step 3.1

Rewrite

315

$315$ as an angle where the values of the six trigonometric functions are known divided by

2

$2$ .

\cos (\frac{630}{2})

$\cos (\frac{630}{2})$

Step 3.2

Apply the cosine half-angle identity

\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}

$\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

\pm \sqrt{\frac{1 + \cos (630)}{2}}

$\pm \sqrt{\frac{1 + \cos (630)}{2}}$

Step 3.3

Change the

\pm

$\pm$ to

+

$+$ because cosine is positive in the fourth quadrant.

\sqrt{\frac{1 + \cos (630)}{2}}

$\sqrt{\frac{1 + \cos (630)}{2}}$

Step 3.4

Simplify

\sqrt{\frac{1 + \cos (630)}{2}}

$\sqrt{\frac{1 + \cos (630)}{2}}$ .

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Step 3.4.1

Remove full rotations of

360

$360$ ° until the angle is between

0

$0$ ° and

360

$360$ °.

\sqrt{\frac{1 + \cos (270)}{2}}

$\sqrt{\frac{1 + \cos (270)}{2}}$

Step 3.4.2

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

\sqrt{\frac{1 - \cos (90)}{2}}

$\sqrt{\frac{1 - \cos (90)}{2}}$

Step 3.4.3

The exact value of

\cos (90)

$\cos (90)$ is

0

$0$ .

\sqrt{\frac{1 - 0}{2}}

$\sqrt{\frac{1 - 0}{2}}$

Step 3.4.4

Multiply

- 1

$- 1$ by

0

$0$ .

\sqrt{\frac{1 + 0}{2}}

$\sqrt{\frac{1 + 0}{2}}$

Step 3.4.5

Add

1

$1$ and

0

$0$ .

\sqrt{\frac{1}{2}}

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

Step 3.4.6

Rewrite

\sqrt{\frac{1}{2}}

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

\frac{\sqrt{1}}{\sqrt{2}}

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

\frac{\sqrt{1}}{\sqrt{2}}

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

Step 3.4.7

Any root of

1

$1$ is

1

$1$ .

\frac{1}{\sqrt{2}}

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

Step 3.4.8

Multiply

\frac{1}{\sqrt{2}}

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

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

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

\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}

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

Step 3.4.9

Combine and simplify the denominator.

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Step 3.4.9.1

Multiply

\frac{1}{\sqrt{2}}

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

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

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

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

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

Step 3.4.9.2

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

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

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

Step 3.4.9.3

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

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

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

Step 3.4.9.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 3.4.9.5

Add

1

$1$ and

1

$1$ .

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

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

Step 3.4.9.6

Rewrite

{\sqrt{2}}^{2}

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

2

$2$ .

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Step 3.4.9.6.1

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

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

Step 3.4.9.6.2

Apply the power rule and multiply exponents,

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

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

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

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

Step 3.4.9.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 3.4.9.6.4

Cancel the common factor of

2

$2$ .

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Step 3.4.9.6.4.1

Cancel the common factor.

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

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

Step 3.4.9.6.4.2

Rewrite the expression.

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

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

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

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

Step 3.4.9.6.5

Evaluate the exponent.

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

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

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

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

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

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

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

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

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

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

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

Decimal Form:

0.70710678 \dots

$0.70710678 \dots$