What is the general solution for 0.5cos(x)=cos(x) ?

The general solution for 0.5cos(x)=cos(x) is x= pi/2+2pin,x=(3pi)/2+2pin

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0.5cos(x)=cos(x)

Solution

0.5 cos (x) = cos (x)

Solution

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

0.5 cos (x) = cos (x)

Solve by substitution

0.5 cos (x) = cos (x)

Let:

cos (x) = u

0.5 u = u

0.5 u = u : u = 0

0.5 u = u

Multiply both sides by

10

0.5 u = u

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 u \cdot 10 = u \cdot 10

Refine

5 u = u \cdot 10

5 u = u \cdot 10

Move

u 10

to the left side

5 u = u \cdot 10

Subtract

u 10

from both sides

5 u - u \cdot 10 = u \cdot 10 - u \cdot 10

Simplify

- 5 u = 0

- 5 u = 0

Divide both sides by

- 5

- 5 u = 0

Divide both sides by

- 5

\frac{- 5 u}{- 5} = \frac{0}{- 5}

Simplify

u = 0

u = 0

Substitute back

u = cos (x)

cos (x) = 0

cos (x) = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

General solutions for

cos (x) = 0

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

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Popular Examples

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tan (a) = - \frac{1}{2}

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

8tan^2(θ)-1=7tan(θ)

8 tan^{2} (θ) - 1 = 7 tan (θ)

2sin^{(2)}(x)-3sin(x)+1=0

2 sin^{(2)} (x) - 3 sin (x) + 1 = 0

sin(x+pi/4)=1

sin (x + \frac{π}{4}) = 1

Frequently Asked Questions (FAQ)

What is the general solution for 0.5cos(x)=cos(x) ?
The general solution for 0.5cos(x)=cos(x) is x= pi/2+2pin,x=(3pi)/2+2pin