What is the general solution for (8cos(θ)-4)(5sin(θ)-5)=0 ?

The general solution for (8cos(θ)-4)(5sin(θ)-5)=0 is θ= pi/3+2pin,θ=(5pi)/3+2pin,θ= pi/2+2pin

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Popular >

(8cos(θ)-4)(5sin(θ)-5)=0

Solution

(8 cos (θ) - 4) (5 sin (θ) - 5) = 0

Solution

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn, θ = \frac{π}{2} + 2 πn

Degrees

θ = 6 0^{\circ} + 36 0^{\circ} n, θ = 30 0^{\circ} + 36 0^{\circ} n, θ = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

(8 cos (θ) - 4) (5 sin (θ) - 5) = 0

Solving each part separately

8 cos (θ) - 4 = 0 or 5 sin (θ) - 5 = 0

8 cos (θ) - 4 = 0 : θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

8 cos (θ) - 4 = 0

Move

4

to the right side

8 cos (θ) - 4 = 0

Add

4

to both sides

8 cos (θ) - 4 + 4 = 0 + 4

Simplify

8 cos (θ) = 4

8 cos (θ) = 4

Divide both sides by

8

8 cos (θ) = 4

Divide both sides by

8

\frac{8 cos ( θ )}{8} = \frac{4}{8}

Simplify

cos (θ) = \frac{1}{2}

cos (θ) = \frac{1}{2}

General solutions for

cos (θ) = \frac{1}{2}

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}

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

5 sin (θ) - 5 = 0 : θ = \frac{π}{2} + 2 πn

5 sin (θ) - 5 = 0

Move

5

to the right side

5 sin (θ) - 5 = 0

Add

5

to both sides

5 sin (θ) - 5 + 5 = 0 + 5

Simplify

5 sin (θ) = 5

5 sin (θ) = 5

Divide both sides by

5

5 sin (θ) = 5

Divide both sides by

5

\frac{5 sin ( θ )}{5} = \frac{5}{5}

Simplify

sin (θ) = 1

sin (θ) = 1

General solutions for

sin (θ) = 1

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}

θ = \frac{π}{2} + 2 πn

θ = \frac{π}{2} + 2 πn

Combine all the solutions

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn, θ = \frac{π}{2} + 2 πn

Popular Examples

arctan(x)= pi/3

arctan (x) = \frac{π}{3}

2cos(2x)-2sin(x)=0

2 cos (2 x) - 2 sin (x) = 0

cos(2x)+3cos(x)-1=0

cos (2 x) + 3 cos (x) - 1 = 0

cos(2θ)=-1

cos (2 θ) = - 1

2sin^2(x)-1=sin(x)

2 sin^{2} (x) - 1 = sin (x)

Frequently Asked Questions (FAQ)

What is the general solution for (8cos(θ)-4)(5sin(θ)-5)=0 ?
The general solution for (8cos(θ)-4)(5sin(θ)-5)=0 is θ= pi/3+2pin,θ=(5pi)/3+2pin,θ= pi/2+2pin