Trigonometry Examples

arccos (x) + 2 arcsin (\frac{\sqrt{3}}{2}) = \frac{π}{3}

$\arccos (x) + 2 \arcsin (\frac{\sqrt{3}}{2}) = \frac{π}{3}$

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

The exact value of

arcsin (\frac{\sqrt{3}}{2})

$\arcsin (\frac{\sqrt{3}}{2})$ is

\frac{π}{3}

$\frac{π}{3}$ .

arccos (x) + 2 \frac{π}{3} = \frac{π}{3}

$\arccos (x) + 2 \frac{π}{3} = \frac{π}{3}$

Step 1.1.2

Combine

2

$2$ and

\frac{π}{3}

$\frac{π}{3}$ .

arccos (x) + \frac{2 π}{3} = \frac{π}{3}

$\arccos (x) + \frac{2 π}{3} = \frac{π}{3}$

arccos (x) + \frac{2 π}{3} = \frac{π}{3}

$\arccos (x) + \frac{2 π}{3} = \frac{π}{3}$

arccos (x) + \frac{2 π}{3} = \frac{π}{3}

$\arccos (x) + \frac{2 π}{3} = \frac{π}{3}$

Step 2

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 2.1

Subtract

\frac{2 π}{3}

$\frac{2 π}{3}$ from both sides of the equation.

arccos (x) = \frac{π}{3} - \frac{2 π}{3}

$\arccos (x) = \frac{π}{3} - \frac{2 π}{3}$

Step 2.2

Combine the numerators over the common denominator.

arccos (x) = \frac{π - 2 π}{3}

$\arccos (x) = \frac{π - 2 π}{3}$

Step 2.3

Subtract

2 π

$2 π$ from

π

$π$ .

arccos (x) = \frac{- π}{3}

$\arccos (x) = \frac{- π}{3}$

Step 2.4

Move the negative in front of the fraction.

$\arccos (x) = - \frac{π}{3}$

Step 3

Take the inverse arccosine of both sides of the equation to extract

$x$ from inside the arccosine.

$x = \cos (- \frac{π}{3})$

Step 4

Simplify the right side.

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Step 4.1

Simplify

$\cos (- \frac{π}{3})$ .

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Step 4.1.1

Add full rotations of

$2 π$ until the angle is greater than or equal to

$0$ and less than

$2 π$ .

$x = \cos (\frac{5 π}{3})$

Step 4.1.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$x = \cos (\frac{π}{3})$

Step 4.1.3

The exact value of

$\cos (\frac{π}{3})$ is

$\frac{1}{2}$ .

$x = \frac{1}{2}$

Step 5

Exclude the solutions that do not make

$\arccos (x) + 2 \arcsin (\frac{\sqrt{3}}{2}) = \frac{π}{3}$ true.

$No solution$