Trigonometry Examples

$n - π \sin (15) - 85 \cdot 9.81 \cos (15)$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\sin (15)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 1.1.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$n - π \sin (45 - 30) - 85 \cdot (9.81 \cos (15))$

Step 1.1.2

Separate negation.

$n - π \sin (45 - (30)) - 85 \cdot (9.81 \cos (15))$

Step 1.1.3

Apply the difference of angles identity.

$n - π (\sin (45) \cos (30) - \cos (45) \sin (30)) - 85 \cdot (9.81 \cos (15))$

Step 1.1.4

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

$n - π (\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30)) - 85 \cdot (9.81 \cos (15))$

Step 1.1.5

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

$n - π (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30)) - 85 \cdot (9.81 \cos (15))$

Step 1.1.6

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

$n - π (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30)) - 85 \cdot (9.81 \cos (15))$

Step 1.1.7

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$n - π (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - 85 \cdot (9.81 \cos (15))$

Step 1.1.8

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 1.1.8.1

Simplify each term.

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Step 1.1.8.1.1

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

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Step 1.1.8.1.1.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{2}$ .

$n - π (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - 85 \cdot (9.81 \cos (15))$

Step 1.1.8.1.1.2

Combine using the product rule for radicals.

$n - π (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - 85 \cdot (9.81 \cos (15))$

Step 1.1.8.1.1.3

Multiply $2$ by $3$ .

$n - π (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - 85 \cdot (9.81 \cos (15))$

Step 1.1.8.1.1.4

Multiply $2$ by $2$ .

$n - π (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - 85 \cdot (9.81 \cos (15))$

Step 1.1.8.1.2

Multiply $- \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 1.1.8.1.2.1

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

$n - π (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}) - 85 \cdot (9.81 \cos (15))$

Step 1.1.8.1.2.2

Multiply $2$ by $2$ .

$n - π (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}) - 85 \cdot (9.81 \cos (15))$

Step 1.1.8.2

Combine the numerators over the common denominator.

$n - π \frac{\sqrt{6} - \sqrt{2}}{4} - 85 \cdot (9.81 \cos (15))$

Step 1.2

Combine $\frac{\sqrt{6} - \sqrt{2}}{4}$ and $π$ .

$n - \frac{(\sqrt{6} - \sqrt{2}) π}{4} - 85 \cdot (9.81 \cos (15))$

Step 1.3

The exact value of $\cos (15)$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

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Step 1.3.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$n - \frac{(\sqrt{6} - \sqrt{2}) π}{4} - 85 \cdot (9.81 \cos (45 - 30))$

Step 1.3.2

Separate negation.

$n - \frac{(\sqrt{6} - \sqrt{2}) π}{4} - 85 \cdot (9.81 \cos (45 - (30)))$

Step 1.3.3

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

$n - \frac{(\sqrt{6} - \sqrt{2}) π}{4} - 85 \cdot (9.81 (\cos (45) \cos (30) + \sin (45) \sin (30)))$

Step 1.3.4

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

$n - \frac{(\sqrt{6} - \sqrt{2}) π}{4} - 85 \cdot (9.81 (\frac{\sqrt{2}}{2} \cos (30) + \sin (45) \sin (30)))$

Step 1.3.5

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

$n - \frac{(\sqrt{6} - \sqrt{2}) π}{4} - 85 \cdot (9.81 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30)))$

Step 1.3.6

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

$n - \frac{(\sqrt{6} - \sqrt{2}) π}{4} - 85 \cdot (9.81 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (30)))$

Step 1.3.7

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$n - \frac{(\sqrt{6} - \sqrt{2}) π}{4} - 85 \cdot (9.81 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}))$

Step 1.3.8

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 1.3.8.1

Simplify each term.

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Step 1.3.8.1.1

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

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Step 1.3.8.1.1.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{2}$ .

$n - \frac{(\sqrt{6} - \sqrt{2}) π}{4} - 85 \cdot (9.81 (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}))$

Step 1.3.8.1.1.2

Combine using the product rule for radicals.

$n - \frac{(\sqrt{6} - \sqrt{2}) π}{4} - 85 \cdot (9.81 (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}))$

Step 1.3.8.1.1.3

Multiply $2$ by $3$ .

$n - \frac{(\sqrt{6} - \sqrt{2}) π}{4} - 85 \cdot (9.81 (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}))$

Step 1.3.8.1.1.4

Multiply $2$ by $2$ .

$n - \frac{(\sqrt{6} - \sqrt{2}) π}{4} - 85 \cdot (9.81 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}))$

Step 1.3.8.1.2

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

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Step 1.3.8.1.2.1

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

$n - \frac{(\sqrt{6} - \sqrt{2}) π}{4} - 85 \cdot (9.81 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2}))$

Step 1.3.8.1.2.2

Multiply $2$ by $2$ .

$n - \frac{(\sqrt{6} - \sqrt{2}) π}{4} - 85 \cdot (9.81 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}))$

Step 1.3.8.2

Combine the numerators over the common denominator.

$n - \frac{(\sqrt{6} - \sqrt{2}) π}{4} - 85 \cdot (9.81 \frac{\sqrt{6} + \sqrt{2}}{4})$

Step 1.4

Multiply $9.81 \frac{\sqrt{6} + \sqrt{2}}{4}$ .

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Step 1.4.1

Combine $9.81$ and $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

$n - \frac{(\sqrt{6} - \sqrt{2}) π}{4} - 85 \cdot \frac{9.81 (\sqrt{6} + \sqrt{2})}{4}$

Step 1.4.2

Multiply $9.81$ by $\sqrt{6} + \sqrt{2}$ .

$n - \frac{(\sqrt{6} - \sqrt{2}) π}{4} - 85 \cdot \frac{37.90292942}{4}$

Step 1.5

Divide $37.90292942$ by $4$ .

$n - \frac{(\sqrt{6} - \sqrt{2}) π}{4} - 85 \cdot 9.47573235$

Step 1.6

Multiply $- 85$ by $9.47573235$ .

$n - \frac{(\sqrt{6} - \sqrt{2}) π}{4} - 805.43725025$

Step 2

Find the common denominator.

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Step 2.1

Write $n$ as a fraction with denominator $1$ .

$\frac{n}{1} - \frac{(\sqrt{6} - \sqrt{2}) π}{4} - 805.43725025$

Step 2.2

Multiply $\frac{n}{1}$ by $\frac{4}{4}$ .

$\frac{n}{1} \cdot \frac{4}{4} - \frac{(\sqrt{6} - \sqrt{2}) π}{4} - 805.43725025$

Step 2.3

Multiply $\frac{n}{1}$ by $\frac{4}{4}$ .

$\frac{n \cdot 4}{4} - \frac{(\sqrt{6} - \sqrt{2}) π}{4} - 805.43725025$

Step 2.4

Write $- 805.43725025$ as a fraction with denominator $1$ .

$\frac{n \cdot 4}{4} - \frac{(\sqrt{6} - \sqrt{2}) π}{4} + \frac{- 805.43725025}{1}$

Step 2.5

Multiply $\frac{- 805.43725025}{1}$ by $\frac{4}{4}$ .

$\frac{n \cdot 4}{4} - \frac{(\sqrt{6} - \sqrt{2}) π}{4} + \frac{- 805.43725025}{1} \cdot \frac{4}{4}$

Step 2.6

Multiply $\frac{- 805.43725025}{1}$ by $\frac{4}{4}$ .

$\frac{n \cdot 4}{4} - \frac{(\sqrt{6} - \sqrt{2}) π}{4} + \frac{- 805.43725025 \cdot 4}{4}$

Step 3

Combine the numerators over the common denominator.

$\frac{n \cdot 4 - (\sqrt{6} - \sqrt{2}) π - 805.43725025 \cdot 4}{4}$

Step 4

Simplify each term.

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Step 4.1

Move $4$ to the left of $n$ .

$\frac{4 \cdot n - (\sqrt{6} - \sqrt{2}) π - 805.43725025 \cdot 4}{4}$

Step 4.2

Apply the distributive property.

$\frac{4 n + (- \sqrt{6} - - \sqrt{2}) π - 805.43725025 \cdot 4}{4}$

Step 4.3

Multiply $- - \sqrt{2}$ .

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Step 4.3.1

Multiply $- 1$ by $- 1$ .

$\frac{4 n + (- \sqrt{6} + 1 \sqrt{2}) π - 805.43725025 \cdot 4}{4}$

Step 4.3.2

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

$\frac{4 n + (- \sqrt{6} + \sqrt{2}) π - 805.43725025 \cdot 4}{4}$

Step 4.4

Apply the distributive property.

$\frac{4 n - \sqrt{6} π + \sqrt{2} π - 805.43725025 \cdot 4}{4}$

Step 4.5

Multiply $- 805.43725025$ by $4$ .

$\frac{4 n - \sqrt{6} π + \sqrt{2} π - 3221.749001}{4}$

Step 5

Simplify by adding terms.

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Step 5.1

Subtract $3221.749001$ from $- \sqrt{6} π$ .

$\frac{4 n - 3229.44429998 + \sqrt{2} π}{4}$

Step 5.2

Add $- 3229.44429998$ and $\sqrt{2} π$ .

$\frac{4 n - 3225.00141704}{4}$