What is the general solution for arcsin(x)-arccos(-1/2)= pi/6 ?

The general solution for arcsin(x)-arccos(-1/2)= pi/6 is No Solution for x\in\mathbb{R}

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arcsin(x)-arccos(-1/2)= pi/6

Solution

arcsin (x) - arccos (- \frac{1}{2}) = \frac{π}{6}

Solution

No Solution for x \in R

Solution steps

arcsin (x) - arccos (- \frac{1}{2}) = \frac{π}{6}

arccos (- \frac{1}{2}) = \frac{2 π}{3}

arccos (- \frac{1}{2})

Use the following trivial identity:

arccos (- \frac{1}{2}) = \frac{2 π}{3}

arccos (- \frac{1}{2})

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= \frac{2 π}{3}

= \frac{2 π}{3}

arcsin (x) - \frac{2 π}{3} = \frac{π}{6}

Move

\frac{2 π}{3}

to the right side

arcsin (x) - \frac{2 π}{3} = \frac{π}{6}

Add

\frac{2 π}{3}

to both sides

arcsin (x) - \frac{2 π}{3} + \frac{2 π}{3} = \frac{π}{6} + \frac{2 π}{3}

Simplify

arcsin (x) - \frac{2 π}{3} + \frac{2 π}{3} = \frac{π}{6} + \frac{2 π}{3}

Simplify

arcsin (x) - \frac{2 π}{3} + \frac{2 π}{3} : arcsin (x)

arcsin (x) - \frac{2 π}{3} + \frac{2 π}{3}

Add similar elements:

- \frac{2 π}{3} + \frac{2 π}{3} = 0

= arcsin (x)

Simplify

\frac{π}{6} + \frac{2 π}{3} : \frac{5 π}{6}

\frac{π}{6} + \frac{2 π}{3}

Least Common Multiplier of

6, 3 : 6

6, 3

Least Common Multiplier (LCM)

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Multiply each factor the greatest number of times it occurs in either

6

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{2 π}{3} :

multiply the denominator and numerator by

2

\frac{2 π}{3} = \frac{2 π 2}{3 \cdot 2} = \frac{4 π}{6}

= \frac{π}{6} + \frac{4 π}{6}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{π + 4 π}{6}

Add similar elements:

π + 4 π = 5 π

= \frac{5 π}{6}

arcsin (x) = \frac{5 π}{6}

arcsin (x) = \frac{5 π}{6}

arcsin (x) = \frac{5 π}{6}

- \frac{π}{2} \leq arcsin (x) \leq \frac{π}{2}

No Solution for x \in R

Popular Examples

tan^2(x)-tan(x)-2=0

tan^{2} (x) - tan (x) - 2 = 0

tan^3(x)=3tan(x)

tan^{3} (x) = 3 tan (x)

sin(2x)=-cos(2x)

sin (2 x) = - cos (2 x)

3sin(4x)+5=6

3 sin (4 x) + 5 = 6

4sin(x)+3=0

4 sin (x) + 3 = 0

Frequently Asked Questions (FAQ)

What is the general solution for arcsin(x)-arccos(-1/2)= pi/6 ?
The general solution for arcsin(x)-arccos(-1/2)= pi/6 is No Solution for x\in\mathbb{R}