What is the general solution for sin(4x+10)=sin(90) ?

The general solution for sin(4x+10)=sin(90) is x=-5/2+22.5+(180n)/2

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sin(4x+10)=sin(90)

Solution

sin (4 x + 10) = sin (9 0^{\circ})

Solution

x = - \frac{5}{2} + 22. 5^{\circ} + \frac{18 0 ^{\circ} n}{2}

Radians

x = - \frac{5}{2} + \frac{π}{8} + \frac{π}{2} n

Solution steps

sin (4 x + 10) = sin (9 0^{\circ})

sin (9 0^{\circ}) = 1

sin (9 0^{\circ})

Use the following trivial identity:

sin (9 0^{\circ}) = 1

sin (9 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 1

= 1

sin (4 x + 10) = 1

General solutions for

sin (4 x + 10) = 1

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

4 x + 10 = 9 0^{\circ} + 36 0^{\circ} n

4 x + 10 = 9 0^{\circ} + 36 0^{\circ} n

Solve

4 x + 10 = 9 0^{\circ} + 36 0^{\circ} n : x = - \frac{5}{2} + 22. 5^{\circ} + \frac{18 0 ^{\circ} n}{2}

4 x + 10 = 9 0^{\circ} + 36 0^{\circ} n

Move

10

to the right side

4 x + 10 = 9 0^{\circ} + 36 0^{\circ} n

Subtract

10

from both sides

4 x + 10 - 10 = 9 0^{\circ} + 36 0^{\circ} n - 10

Simplify

4 x = 9 0^{\circ} + 36 0^{\circ} n - 10

4 x = 9 0^{\circ} + 36 0^{\circ} n - 10

Divide both sides by

4

4 x = 9 0^{\circ} + 36 0^{\circ} n - 10

Divide both sides by

4

\frac{4 x}{4} = \frac{9 0 ^{\circ}}{4} + \frac{36 0 ^{\circ} n}{4} - \frac{10}{4}

Simplify

\frac{4 x}{4} = \frac{9 0 ^{\circ}}{4} + \frac{36 0 ^{\circ} n}{4} - \frac{10}{4}

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{9 0 ^{\circ}}{4} + \frac{36 0 ^{\circ} n}{4} - \frac{10}{4} : - \frac{5}{2} + 22. 5^{\circ} + \frac{18 0 ^{\circ} n}{2}

\frac{9 0 ^{\circ}}{4} + \frac{36 0 ^{\circ} n}{4} - \frac{10}{4}

Group like terms

= - \frac{10}{4} + \frac{36 0 ^{\circ} n}{4} + \frac{9 0 ^{\circ}}{4}

\frac{10}{4} = \frac{5}{2}

\frac{10}{4}

Cancel the common factor:

2

= \frac{5}{2}

\frac{36 0 ^{\circ} n}{4} = \frac{18 0 ^{\circ} n}{2}

\frac{36 0 ^{\circ} n}{4}

Cancel the common factor:

2

= \frac{18 0 ^{\circ} n}{2}

\frac{9 0 ^{\circ}}{4} = 22. 5^{\circ}

\frac{9 0 ^{\circ}}{4}

Apply the fraction rule:

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

= \frac{18 0 ^{\circ}}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= 22. 5^{\circ}

= - \frac{5}{2} + \frac{18 0 ^{\circ} n}{2} + 22. 5^{\circ}

Group like terms

= - \frac{5}{2} + 22. 5^{\circ} + \frac{18 0 ^{\circ} n}{2}

x = - \frac{5}{2} + 22. 5^{\circ} + \frac{18 0 ^{\circ} n}{2}

x = - \frac{5}{2} + 22. 5^{\circ} + \frac{18 0 ^{\circ} n}{2}

x = - \frac{5}{2} + 22. 5^{\circ} + \frac{18 0 ^{\circ} n}{2}

x = - \frac{5}{2} + 22. 5^{\circ} + \frac{18 0 ^{\circ} n}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(4x+10)=sin(90) ?
The general solution for sin(4x+10)=sin(90) is x=-5/2+22.5+(180n)/2