What is the general solution for cos^2(θ)+sqrt(3)sin(θ)cos(θ)=1 ?

The general solution for cos^2(θ)+sqrt(3)sin(θ)cos(θ)=1 is θ=2pin,θ=pi+2pin,θ= pi/3+pin

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Popular >

cos^2(θ)+sqrt(3)sin(θ)cos(θ)=1

Solution

cos^{2} (θ) + 3 sin (θ) cos (θ) = 1

Solution

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

Degrees

θ = 0^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n, θ = 6 0^{\circ} + 18 0^{\circ} n

Solution steps

cos^{2} (θ) + 3 sin (θ) cos (θ) = 1

Subtract

1

from both sides

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

Rewrite using trig identities

- 1 + cos^{2} (θ) + cos (θ) sin (θ) 3

Use the Pythagorean identity:

1 = cos^{2} (x) + sin^{2} (x)

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

= 3 cos (θ) sin (θ) - sin^{2} (θ)

- sin^{2} (θ) + cos (θ) sin (θ) 3 = 0

Factor

- sin^{2} (θ) + cos (θ) sin (θ) 3 : sin (θ) (- sin (θ) + 3 cos (θ))

- sin^{2} (θ) + cos (θ) sin (θ) 3

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

sin^{2} (θ) = sin (θ) sin (θ)

= - sin (θ) sin (θ) + cos (θ) sin (θ) 3

Factor out common term

sin (θ)

= sin (θ) (- sin (θ) + cos (θ) 3)

sin (θ) (- sin (θ) + 3 cos (θ)) = 0

Solving each part separately

sin (θ) = 0 or - sin (θ) + 3 cos (θ) = 0

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

sin (θ) = 0

General solutions for

sin (θ) = 0

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 = 0 + 2 πn, θ = π + 2 πn

θ = 0 + 2 πn, θ = π + 2 πn

Solve

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn, θ = π + 2 πn

- sin (θ) + 3 cos (θ) = 0 : θ = \frac{π}{3} + πn

- sin (θ) + 3 cos (θ) = 0

Rewrite using trig identities

- sin (θ) + 3 cos (θ) = 0

Divide both sides by

cos (θ), cos (θ) \neq = 0

\frac{- sin ( θ ) + 3 cos ( θ )}{cos ( θ )} = \frac{0}{cos ( θ )}

Simplify

- \frac{sin ( θ )}{cos ( θ )} + 3 = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

- tan (θ) + 3 = 0

- tan (θ) + 3 = 0

Move

3

to the right side

- tan (θ) + 3 = 0

Subtract

3

from both sides

- tan (θ) + 3 - 3 = 0 - 3

Simplify

- tan (θ) = - 3

- tan (θ) = - 3

Divide both sides by

- 1

- tan (θ) = - 3

Divide both sides by

- 1

\frac{- tan ( θ )}{- 1} = \frac{- 3}{- 1}

Simplify

tan (θ) = 3

tan (θ) = 3

General solutions for

tan (θ) = 3

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

θ = \frac{π}{3} + πn

θ = \frac{π}{3} + πn

Combine all the solutions

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

Popular Examples

sin((-pi)/6+3x)= 1/(sqrt(2))

sin (\frac{- π}{6} + 3 x) = \frac{1}{2}

tan(a)=-3/4

tan (a) = - \frac{3}{4}

csc(θ)=-3/5 ,cos(θ)

csc (θ) = - \frac{3}{5}, cos (θ)

cos(2x)=0.2

cos (2 x) = 0.2

cos(2x)=0.6

cos (2 x) = 0.6

Frequently Asked Questions (FAQ)

What is the general solution for cos^2(θ)+sqrt(3)sin(θ)cos(θ)=1 ?
The general solution for cos^2(θ)+sqrt(3)sin(θ)cos(θ)=1 is θ=2pin,θ=pi+2pin,θ= pi/3+pin