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Popular Trigonometry >

1sin(x)=1.5sin(x/2)

Solution

1 \cdot sin (x) = 1.5 \cdot sin (\frac{x}{2})

Solution

x = 4 πn, x = 2 π (2 n + 1), x = \frac{0.72273 \dots + 2 πn}{0.5}, x = \frac{2 π - 0.72273 \dots + 2 πn}{0.5}

Degrees

x = 0^{\circ} + 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n, x = 82.81924 \dots^{\circ} + 72 0^{\circ} n, x = 637.18075 \dots^{\circ} + 72 0^{\circ} n

Solution steps

1 \cdot sin (x) = 1.5 sin (\frac{x}{2})

Subtract

1.5 sin (\frac{x}{2})

from both sides

sin (x) - 1.5 sin (0.5 x) = 0

Let:

u = 0.5 x

sin (2 u) - 1.5 sin (u) = 0

Rewrite using trig identities

sin (2 u) - 1.5 sin (u)

Use the Double Angle identity:

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

= 2 sin (u) cos (u) - 1.5 sin (u)

- 1.5 sin (u) + 2 cos (u) sin (u) = 0

Factor

- 1.5 sin (u) + 2 cos (u) sin (u) : sin (u) (2 cos (u) - 1.5)

- 1.5 sin (u) + 2 cos (u) sin (u)

Factor out common term

sin (u)

= sin (u) (- 1.5 + 2 cos (u))

sin (u) (2 cos (u) - 1.5) = 0

Solving each part separately

sin (u) = 0 or 2 cos (u) - 1.5 = 0

sin (u) = 0 : u = 2 πn, u = π + 2 πn

sin (u) = 0

General solutions for

sin (u) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

u = 0 + 2 πn, u = π + 2 πn

u = 0 + 2 πn, u = π + 2 πn

Solve

u = 0 + 2 πn : u = 2 πn

u = 0 + 2 πn

0 + 2 πn = 2 πn

u = 2 πn

u = 2 πn, u = π + 2 πn

2 cos (u) - 1.5 = 0 : u = arccos (0.75) + 2 πn, u = 2 π - arccos (0.75) + 2 πn

2 cos (u) - 1.5 = 0

Move

1.5

to the right side

2 cos (u) - 1.5 = 0

Add

1.5

to both sides

2 cos (u) - 1.5 + 1.5 = 0 + 1.5

Simplify

2 cos (u) = 1.5

2 cos (u) = 1.5

Divide both sides by

2

2 cos (u) = 1.5

Divide both sides by

2

\frac{2 cos ( u )}{2} = \frac{1.5}{2}

Simplify

cos (u) = 0.75

cos (u) = 0.75

Apply trig inverse properties

cos (u) = 0.75

General solutions for

cos (u) = 0.75

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

u = arccos (0.75) + 2 πn, u = 2 π - arccos (0.75) + 2 πn

u = arccos (0.75) + 2 πn, u = 2 π - arccos (0.75) + 2 πn

Combine all the solutions

u = 2 πn, u = π + 2 πn, u = arccos (0.75) + 2 πn, u = 2 π - arccos (0.75) + 2 πn

Substitute back

u = 0.5 x

0.5 x = 2 πn : x = 4 πn

0.5 x = 2 πn

Multiply both sides by

10

0.5 x = 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

0.5 x \cdot 10 = 2 πn \cdot 10

Refine

5 x = 20 πn

5 x = 20 πn

Divide both sides by

5

5 x = 20 πn

Divide both sides by

5

\frac{5 x}{5} = \frac{20 πn}{5}

Simplify

x = 4 πn

x = 4 πn

0.5 x = π + 2 πn : x = 2 π (2 n + 1)

0.5 x = π + 2 πn

Multiply both sides by

10

0.5 x = π + 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

0.5 x \cdot 10 = π 10 + 2 πn \cdot 10

Refine

5 x = 10 π + 20 πn

5 x = 10 π + 20 πn

Divide both sides by

5

5 x = 10 π + 20 πn

Divide both sides by

5

\frac{5 x}{5} = \frac{10 π}{5} + \frac{20 πn}{5}

Simplify

\frac{5 x}{5} = \frac{10 π}{5} + \frac{20 πn}{5}

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{10 π}{5} + \frac{20 πn}{5} : 2 π (2 n + 1)

\frac{10 π}{5} + \frac{20 πn}{5}

Apply rule

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

= \frac{10 π + 20 πn}{5}

Factor

10 π + 20 πn : 10 π (1 + 2 n)

10 π + 20 πn

Rewrite as

= 1 \cdot 10 π + 2 \cdot 10 πn

Factor out common term

10 π

= 10 π (1 + 2 n)

= \frac{10 π ( 1 + 2 n )}{5}

Divide the numbers:

\frac{10}{5} = 2

= 2 π (2 n + 1)

x = 2 π (2 n + 1)

x = 2 π (2 n + 1)

x = 2 π (2 n + 1)

0.5 x = arccos (0.75) + 2 πn : x = \frac{arccos ( 0.75 ) + 2 πn}{0.5}

0.5 x = arccos (0.75) + 2 πn

Divide both sides by

0.5

0.5 x = arccos (0.75) + 2 πn

Divide both sides by

0.5

\frac{0.5 x}{0.5} = \frac{arccos ( 0.75 )}{0.5} + \frac{2 πn}{0.5}

Simplify

\frac{0.5 x}{0.5} = \frac{arccos ( 0.75 )}{0.5} + \frac{2 πn}{0.5}

Simplify

\frac{0.5 x}{0.5} : x

\frac{0.5 x}{0.5}

Cancel the common factor:

0.5

= x

Simplify

\frac{arccos ( 0.75 )}{0.5} + \frac{2 πn}{0.5} : \frac{arccos ( 0.75 ) + 2 πn}{0.5}

\frac{arccos ( 0.75 )}{0.5} + \frac{2 πn}{0.5}

Apply rule

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

= \frac{arccos ( 0.75 ) + 2 πn}{0.5}

x = \frac{arccos ( 0.75 ) + 2 πn}{0.5}

x = \frac{arccos ( 0.75 ) + 2 πn}{0.5}

x = \frac{arccos ( 0.75 ) + 2 πn}{0.5}

0.5 x = 2 π - arccos (0.75) + 2 πn : x = \frac{2 π - arccos ( 0.75 ) + 2 πn}{0.5}

0.5 x = 2 π - arccos (0.75) + 2 πn

Divide both sides by

0.5

0.5 x = 2 π - arccos (0.75) + 2 πn

Divide both sides by

0.5

\frac{0.5 x}{0.5} = \frac{2 π}{0.5} - \frac{arccos ( 0.75 )}{0.5} + \frac{2 πn}{0.5}

Simplify

\frac{0.5 x}{0.5} = \frac{2 π}{0.5} - \frac{arccos ( 0.75 )}{0.5} + \frac{2 πn}{0.5}

Simplify

\frac{0.5 x}{0.5} : x

\frac{0.5 x}{0.5}

Cancel the common factor:

0.5

= x

Simplify

\frac{2 π}{0.5} - \frac{arccos ( 0.75 )}{0.5} + \frac{2 πn}{0.5} : \frac{2 π - arccos ( 0.75 ) + 2 πn}{0.5}

\frac{2 π}{0.5} - \frac{arccos ( 0.75 )}{0.5} + \frac{2 πn}{0.5}

Apply rule

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

= \frac{2 π - arccos ( 0.75 ) + 2 πn}{0.5}

x = \frac{2 π - arccos ( 0.75 ) + 2 πn}{0.5}

x = \frac{2 π - arccos ( 0.75 ) + 2 πn}{0.5}

x = \frac{2 π - arccos ( 0.75 ) + 2 πn}{0.5}

x = 4 πn, x = 2 π (2 n + 1), x = \frac{arccos ( 0.75 ) + 2 πn}{0.5}, x = \frac{2 π - arccos ( 0.75 ) + 2 πn}{0.5}

Show solutions in decimal form

x = 4 πn, x = 2 π (2 n + 1), x = \frac{0.72273 \dots + 2 πn}{0.5}, x = \frac{2 π - 0.72273 \dots + 2 πn}{0.5}

Graph

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1*sin(x)=1.5*sin(x/2)

Solution

Solution

Graph

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1sin(x)=1.5sin(x/2)