Trigonometry Examples

$\sin (70) \cos (x) - \cos (70) \sin (x) = \frac{\sqrt{3}}{2}$

Step 1

Simplify each term.

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Step 1.1

Evaluate $\sin (70)$ .

$0.93969262 \cos (x) - \cos (70) \sin (x) = \frac{\sqrt{3}}{2}$

Step 1.2

Evaluate $\cos (70)$ .

$0.93969262 \cos (x) - 1 \cdot (0.34202014 \sin (x)) = \frac{\sqrt{3}}{2}$

Step 1.3

Multiply $- 1$ by $0.34202014$ .

$0.93969262 \cos (x) - 0.34202014 \sin (x) = \frac{\sqrt{3}}{2}$

Step 2

Use the identity to solve the equation. In this identity, $θ$ represents the angle created by plotting point $(a, b)$ on a graph and therefore can be found using $θ = \tan^{-1} (\frac{b}{a})$ .

$a \sin (x) + b \cos (x) = R \sin (x + θ)$ where $R = \sqrt{a^{2} + b^{2}}$ and $θ = \tan^{-1} (\frac{b}{a})$

Step 3

Set up the equation to find the value of $θ$ .

$\tan^{-1} (\frac{0.93969262}{- 0.34202014})$

Step 4

Take the inverse tangent to solve the equation for $θ$ .

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Step 4.1

Divide $0.93969262$ by $- 0.34202014$ .

$θ = \tan^{-1} (- 2.74747741)$

Step 4.2

Evaluate $\tan^{-1} (- 2.74747741)$ .

$θ = - 70$

Step 5

Solve to find the value of $R$ .

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Step 5.1

Raise $- 0.34202014$ to the power of $2$ .

$R = \sqrt{0.11697777 + {(0.93969262)}^{2}}$

Step 5.2

Raise $0.93969262$ to the power of $2$ .

$R = \sqrt{0.11697777 + 0.88302222}$

Step 5.3

Add $0.11697777$ and $0.88302222$ .

$R = \sqrt{1}$

Step 6

Substitute the known values into the equation.

$(\sqrt{1}) \sin (x - 70) = \frac{\sqrt{3}}{2}$

Step 7

Divide each term in $\sqrt{1} \sin (x - 70) = \frac{\sqrt{3}}{2}$ by $\sqrt{1}$ and simplify.

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Step 7.1

Divide each term in $\sqrt{1} \sin (x - 70) = \frac{\sqrt{3}}{2}$ by $\sqrt{1}$ .

$\frac{\sqrt{1} \sin (x - 70)}{\sqrt{1}} = \frac{\frac{\sqrt{3}}{2}}{\sqrt{1}}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $\sqrt{1}$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{\sqrt{1} \sin (x - 70)}{\sqrt{1}} = \frac{\frac{\sqrt{3}}{2}}{\sqrt{1}}$

Step 7.2.1.2

Divide $\sin (x - 70)$ by $1$ .

$\sin (x - 70) = \frac{\frac{\sqrt{3}}{2}}{\sqrt{1}}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Multiply the numerator by the reciprocal of the denominator.

$\sin (x - 70) = \frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{1}}$

Step 7.3.2

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

$\sin (x - 70) = \frac{\sqrt{3}}{2} (\frac{1}{\sqrt{1}} \cdot \frac{\sqrt{1}}{\sqrt{1}})$

Step 7.3.3

Combine and simplify the denominator.

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Step 7.3.3.1

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

$\sin (x - 70) = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{1}}{\sqrt{1} \sqrt{1}}$

Step 7.3.3.2

Raise $\sqrt{1}$ to the power of $1$ .

$\sin (x - 70) = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{1}}{{\sqrt{1}}^{1} \sqrt{1}}$

Step 7.3.3.3

Raise $\sqrt{1}$ to the power of $1$ .

$\sin (x - 70) = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{1}}{{\sqrt{1}}^{1} {\sqrt{1}}^{1}}$

Step 7.3.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (x - 70) = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{1}}{{\sqrt{1}}^{1 + 1}}$

Step 7.3.3.5

Add $1$ and $1$ .

$\sin (x - 70) = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{1}}{{\sqrt{1}}^{2}}$

Step 7.3.3.6

Rewrite ${\sqrt{1}}^{2}$ as $1$ .

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Step 7.3.3.6.1

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

$\sin (x - 70) = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{1}}{{(1^{\frac{1}{2}})}^{2}}$

Step 7.3.3.6.2

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

$\sin (x - 70) = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{1}}{1^{\frac{1}{2} \cdot 2}}$

Step 7.3.3.6.3

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

$\sin (x - 70) = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{1}}{1^{\frac{2}{2}}}$

Step 7.3.3.6.4

Cancel the common factor of $2$ .

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Step 7.3.3.6.4.1

Cancel the common factor.

$\sin (x - 70) = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{1}}{1^{\frac{2}{2}}}$

Step 7.3.3.6.4.2

Rewrite the expression.

$\sin (x - 70) = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{1}}{1^{1}}$

Step 7.3.3.6.5

Evaluate the exponent.

$\sin (x - 70) = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{1}}{1}$

Step 7.3.4

Evaluate the root.

$\sin (x - 70) = \frac{\sqrt{3}}{2} \cdot \frac{1}{1}$

Step 7.3.5

Divide $1$ by $1$ .

$\sin (x - 70) = \frac{\sqrt{3}}{2} \cdot 1$

Step 7.3.6

Multiply $\frac{\sqrt{3}}{2}$ by $1$ .

$\sin (x - 70) = \frac{\sqrt{3}}{2}$

Step 8

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$x - 70 = \sin^{-1} (\frac{\sqrt{3}}{2})$

Step 9

Simplify the right side.

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Step 9.1

The exact value of $\sin^{-1} (\frac{\sqrt{3}}{2})$ is $60$ .

$x - 70 = 60$

Step 10

Move all terms not containing $x$ to the right side of the equation.

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Step 10.1

Add $70$ to both sides of the equation.

$x = 60 + 70$

Step 10.2

Add $60$ and $70$ .

$x = 130$

Step 11

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the second quadrant.

$x - 70 = 180 - 60$

Step 12

Solve for $x$ .

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Step 12.1

Subtract $60$ from $180$ .

$x - 70 = 120$

Step 12.2

Move all terms not containing $x$ to the right side of the equation.

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Step 12.2.1

Add $70$ to both sides of the equation.

$x = 120 + 70$

Step 12.2.2

Add $120$ and $70$ .

$x = 190$

Step 13

Find the period of $\sin (x - 70)$ .

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Step 13.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 13.2

Replace $b$ with $1$ in the formula for period.

$\frac{360}{| 1 |}$

Step 13.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{360}{1}$

Step 13.4

Divide $360$ by $1$ .

$360$

Step 14

The period of the $\sin (x - 70)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$x = 130 + 360 n, 190 + 360 n$ , for any integer $n$

Step 15

Exclude the solutions that do not make $\sin (70) \cos (x) - \cos (70) \sin (x) = \frac{\sqrt{3}}{2}$ true.

No solution