Trigonometry Examples

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

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

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

Step 1.1.2

Combine $2$ and $\frac{π}{3}$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Combine the numerators over the common denominator.

$\arccos (x) = \frac{π - 2 π}{3}$

Step 2.3

Subtract $2 π$ from $π$ .

$\arccos (x) = \frac{- π}{3}$

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.

$No solution$