What is the general solution for cos(2x+30)= 1/2 ,0<= x<= 360 ?

Q: What is the general solution for cos(2x+30)= 1/2 ,0<= x<= 360 ?

The general solution for cos(2x+30)= 1/2 ,0<= x<= 360 is x=(-90+5220)/6

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cos(2x+30)= 1/2 ,0<= x<= 360

Solution

cos (2 x + 30) = \frac{1}{2}, 0^{\circ} \leq x \leq 36 0^{\circ}

Solution

x = \frac{- 90 + 522 0 ^{\circ}}{6}

Radians

x = \frac{- 90 + 29 π}{6}

Solution steps

cos (2 x + 30) = \frac{1}{2}, 0^{\circ} \leq x \leq 36 0^{\circ}

General solutions for

cos (2 x + 30) = \frac{1}{2}

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}

2 x + 30 = 6 0^{\circ} + 36 0^{\circ} n, 2 x + 30 = 30 0^{\circ} + 36 0^{\circ} n

2 x + 30 = 6 0^{\circ} + 36 0^{\circ} n, 2 x + 30 = 30 0^{\circ} + 36 0^{\circ} n

Solve

2 x + 30 = 6 0^{\circ} + 36 0^{\circ} n : x = - 15 + 18 0^{\circ} n + 3 0^{\circ}

2 x + 30 = 6 0^{\circ} + 36 0^{\circ} n

Move

30

to the right side

2 x + 30 = 6 0^{\circ} + 36 0^{\circ} n

Subtract

30

from both sides

2 x + 30 - 30 = 6 0^{\circ} + 36 0^{\circ} n - 30

Simplify

2 x = 6 0^{\circ} + 36 0^{\circ} n - 30

2 x = 6 0^{\circ} + 36 0^{\circ} n - 30

Divide both sides by

2

2 x = 6 0^{\circ} + 36 0^{\circ} n - 30

Divide both sides by

2

\frac{2 x}{2} = \frac{6 0 ^{\circ}}{2} + \frac{36 0 ^{\circ} n}{2} - \frac{30}{2}

Simplify

\frac{2 x}{2} = \frac{6 0 ^{\circ}}{2} + \frac{36 0 ^{\circ} n}{2} - \frac{30}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{6 0 ^{\circ}}{2} + \frac{36 0 ^{\circ} n}{2} - \frac{30}{2} : - 15 + 18 0^{\circ} n + 3 0^{\circ}

\frac{6 0 ^{\circ}}{2} + \frac{36 0 ^{\circ} n}{2} - \frac{30}{2}

Group like terms

= - \frac{30}{2} + \frac{36 0 ^{\circ} n}{2} + \frac{6 0 ^{\circ}}{2}

\frac{30}{2} = 15

\frac{30}{2}

Divide the numbers:

\frac{30}{2} = 15

= 15

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

\frac{36 0 ^{\circ} n}{2}

Divide the numbers:

\frac{2}{2} = 1

= 18 0^{\circ} n

\frac{6 0 ^{\circ}}{2} = 3 0^{\circ}

\frac{6 0 ^{\circ}}{2}

Apply the fraction rule:

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

= \frac{18 0 ^{\circ}}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= 3 0^{\circ}

= - 15 + 18 0^{\circ} n + 3 0^{\circ}

x = - 15 + 18 0^{\circ} n + 3 0^{\circ}

x = - 15 + 18 0^{\circ} n + 3 0^{\circ}

x = - 15 + 18 0^{\circ} n + 3 0^{\circ}

Solve

2 x + 30 = 30 0^{\circ} + 36 0^{\circ} n : x = - 15 + 18 0^{\circ} n + 15 0^{\circ}

2 x + 30 = 30 0^{\circ} + 36 0^{\circ} n

Move

30

to the right side

2 x + 30 = 30 0^{\circ} + 36 0^{\circ} n

Subtract

30

from both sides

2 x + 30 - 30 = 30 0^{\circ} + 36 0^{\circ} n - 30

Simplify

2 x = 30 0^{\circ} + 36 0^{\circ} n - 30

2 x = 30 0^{\circ} + 36 0^{\circ} n - 30

Divide both sides by

2

2 x = 30 0^{\circ} + 36 0^{\circ} n - 30

Divide both sides by

2

\frac{2 x}{2} = \frac{30 0 ^{\circ}}{2} + \frac{36 0 ^{\circ} n}{2} - \frac{30}{2}

Simplify

\frac{2 x}{2} = \frac{30 0 ^{\circ}}{2} + \frac{36 0 ^{\circ} n}{2} - \frac{30}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{30 0 ^{\circ}}{2} + \frac{36 0 ^{\circ} n}{2} - \frac{30}{2} : - 15 + 18 0^{\circ} n + 15 0^{\circ}

\frac{30 0 ^{\circ}}{2} + \frac{36 0 ^{\circ} n}{2} - \frac{30}{2}

Group like terms

= - \frac{30}{2} + \frac{36 0 ^{\circ} n}{2} + \frac{30 0 ^{\circ}}{2}

\frac{30}{2} = 15

\frac{30}{2}

Divide the numbers:

\frac{30}{2} = 15

= 15

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

\frac{36 0 ^{\circ} n}{2}

Divide the numbers:

\frac{2}{2} = 1

= 18 0^{\circ} n

\frac{30 0 ^{\circ}}{2} = 15 0^{\circ}

\frac{30 0 ^{\circ}}{2}

Apply the fraction rule:

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

= \frac{90 0 ^{\circ}}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= 15 0^{\circ}

= - 15 + 18 0^{\circ} n + 15 0^{\circ}

x = - 15 + 18 0^{\circ} n + 15 0^{\circ}

x = - 15 + 18 0^{\circ} n + 15 0^{\circ}

x = - 15 + 18 0^{\circ} n + 15 0^{\circ}

x = - 15 + 18 0^{\circ} n + 3 0^{\circ}, x = - 15 + 18 0^{\circ} n + 15 0^{\circ}

Solutions for the range

0 \leq x \leq 36 0^{\circ}

x = \frac{- 90 + 522 0 ^{\circ}}{6}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(2x+30)= 1/2 ,0<= x<= 360 ?
The general solution for cos(2x+30)= 1/2 ,0<= x<= 360 is x=(-90+5220)/6