arccos(2x)=pi/3 — step-by-step solution

1) Let's solve in

ℝ

\arccos (2 x) = \frac{π}{3}

\arccos

takes its values in

[0, π]

, which contains

\frac{π}{3}

\arccos (2 x) = \frac{π}{3}

is equivalent to

2 x = \cos (\frac{π}{3})

ⓘ The cosine is the x-coordinate (horizontal axis) of the point reached on the unit circle after a rotation of

\frac{π}{3}

radians from the horizontal axis pointing to the right, this is a remarkable value:

\cos (\frac{π}{3})

\frac{1}{2}

2 x = \frac{1}{2}

ⓘ Both sides are divided by

2

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

ⓘ Dividing a fraction by

2

is multiplying its denominator by

2

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

\frac{1}{2 \cdot 2}

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

ⅰ

2 \cdot 2

4

Candidates, pending the domain check:

x = \frac{1}{4}

2) Let's evaluate the domain of

\arccos (2 x) = \frac{π}{3}

For the expression

\arccos (2 x)

to exist, you must check:

| 2 x | \leq 1

| A | \leq B

is equivalent to

- B \leq A \leq B

-1 \leq 2 x \leq 1

We divide all three parts by

2

, which is positive

- \frac{1}{2} \leq x \leq \frac{1}{2}

Domain constraints:

x \in [- \frac{1}{2}, \frac{1}{2}]

▶Solution:

x = \frac{1}{4}

3) The numeric value of:

x = \frac{1}{4}

is:

▷

x = 0.25