What is the general solution for sin(5x)-cos(30)=0 ?

The general solution for sin(5x)-cos(30)=0 is x=12+(360n)/5 ,x=24+(360n)/5

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sin(5x)-cos(30)=0

Solution

sin (5 x) - cos (3 0^{\circ}) = 0

Solution

x = 1 2^{\circ} + \frac{36 0 ^{\circ} n}{5}, x = 2 4^{\circ} + \frac{36 0 ^{\circ} n}{5}

Radians

x = \frac{π}{15} + \frac{2 π}{5} n, x = \frac{2 π}{15} + \frac{2 π}{5} n

Solution steps

sin (5 x) - cos (3 0^{\circ}) = 0

cos (3 0^{\circ}) = \frac{3}{2}

cos (3 0^{\circ})

Use the following trivial identity:

cos (3 0^{\circ}) = \frac{3}{2}

cos (3 0^{\circ})

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

= \frac{3}{2}

sin (5 x) - \frac{3}{2} = 0

Move

\frac{3}{2}

to the right side

sin (5 x) - \frac{3}{2} = 0

Add

\frac{3}{2}

to both sides

sin (5 x) - \frac{3}{2} + \frac{3}{2} = 0 + \frac{3}{2}

Simplify

sin (5 x) = \frac{3}{2}

sin (5 x) = \frac{3}{2}

General solutions for

sin (5 x) = \frac{3}{2}

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}

5 x = 6 0^{\circ} + 36 0^{\circ} n, 5 x = 12 0^{\circ} + 36 0^{\circ} n

5 x = 6 0^{\circ} + 36 0^{\circ} n, 5 x = 12 0^{\circ} + 36 0^{\circ} n

Solve

5 x = 6 0^{\circ} + 36 0^{\circ} n : x = 1 2^{\circ} + \frac{36 0 ^{\circ} n}{5}

5 x = 6 0^{\circ} + 36 0^{\circ} n

Divide both sides by

5

5 x = 6 0^{\circ} + 36 0^{\circ} n

Divide both sides by

5

\frac{5 x}{5} = \frac{6 0 ^{\circ}}{5} + \frac{36 0 ^{\circ} n}{5}

Simplify

\frac{5 x}{5} = \frac{6 0 ^{\circ}}{5} + \frac{36 0 ^{\circ} n}{5}

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{6 0 ^{\circ}}{5} + \frac{36 0 ^{\circ} n}{5} : 1 2^{\circ} + \frac{36 0 ^{\circ} n}{5}

\frac{6 0 ^{\circ}}{5} + \frac{36 0 ^{\circ} n}{5}

\frac{6 0 ^{\circ}}{5} = 1 2^{\circ}

\frac{6 0 ^{\circ}}{5}

Apply the fraction rule:

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

= \frac{18 0 ^{\circ}}{3 \cdot 5}

Multiply the numbers:

3 \cdot 5 = 15

= 1 2^{\circ}

= 1 2^{\circ} + \frac{36 0 ^{\circ} n}{5}

x = 1 2^{\circ} + \frac{36 0 ^{\circ} n}{5}

x = 1 2^{\circ} + \frac{36 0 ^{\circ} n}{5}

x = 1 2^{\circ} + \frac{36 0 ^{\circ} n}{5}

Solve

5 x = 12 0^{\circ} + 36 0^{\circ} n : x = 2 4^{\circ} + \frac{36 0 ^{\circ} n}{5}

5 x = 12 0^{\circ} + 36 0^{\circ} n

Divide both sides by

5

5 x = 12 0^{\circ} + 36 0^{\circ} n

Divide both sides by

5

\frac{5 x}{5} = \frac{12 0 ^{\circ}}{5} + \frac{36 0 ^{\circ} n}{5}

Simplify

\frac{5 x}{5} = \frac{12 0 ^{\circ}}{5} + \frac{36 0 ^{\circ} n}{5}

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{12 0 ^{\circ}}{5} + \frac{36 0 ^{\circ} n}{5} : 2 4^{\circ} + \frac{36 0 ^{\circ} n}{5}

\frac{12 0 ^{\circ}}{5} + \frac{36 0 ^{\circ} n}{5}

\frac{12 0 ^{\circ}}{5} = 2 4^{\circ}

\frac{12 0 ^{\circ}}{5}

Apply the fraction rule:

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

= \frac{36 0 ^{\circ}}{3 \cdot 5}

Multiply the numbers:

3 \cdot 5 = 15

= 2 4^{\circ}

= 2 4^{\circ} + \frac{36 0 ^{\circ} n}{5}

x = 2 4^{\circ} + \frac{36 0 ^{\circ} n}{5}

x = 2 4^{\circ} + \frac{36 0 ^{\circ} n}{5}

x = 2 4^{\circ} + \frac{36 0 ^{\circ} n}{5}

x = 1 2^{\circ} + \frac{36 0 ^{\circ} n}{5}, x = 2 4^{\circ} + \frac{36 0 ^{\circ} n}{5}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(5x)-cos(30)=0 ?
The general solution for sin(5x)-cos(30)=0 is x=12+(360n)/5 ,x=24+(360n)/5