What is the general solution for sin^2(a)=3cos^2(a) ?

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

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Popular >

sin^2(a)=3cos^2(a)

Solution

sin^{2} (a) = 3 cos^{2} (a)

Solution

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

Degrees

a = 12 0^{\circ} + 18 0^{\circ} n, a = 6 0^{\circ} + 18 0^{\circ} n

Solution steps

sin^{2} (a) = 3 cos^{2} (a)

Subtract

3 cos^{2} (a)

from both sides

sin^{2} (a) - 3 cos^{2} (a) = 0

Factor

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

sin^{2} (a) - 3 cos^{2} (a)

Rewrite

sin^{2} (a) - 3 cos^{2} (a)

sin^{2} (a) - (3 cos (a))^{2}

sin^{2} (a) - 3 cos^{2} (a)

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

= sin^{2} (a) - (3)^{2} cos^{2} (a)

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(3)^{2} cos^{2} (a) = (3 cos (a))^{2}

= sin^{2} (a) - (3 cos (a))^{2}

= sin^{2} (a) - (3 cos (a))^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

sin^{2} (a) - (3 cos (a))^{2} = (sin (a) + 3 cos (a)) (sin (a) - 3 cos (a))

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

(sin (a) + 3 cos (a)) (sin (a) - 3 cos (a)) = 0

Solving each part separately

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

sin (a) + 3 cos (a) = 0 : a = \frac{2 π}{3} + πn

sin (a) + 3 cos (a) = 0

Rewrite using trig identities

sin (a) + 3 cos (a) = 0

Divide both sides by

cos (a), cos (a) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

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

tan (a) + 3 = 0

tan (a) + 3 = 0

Move

3

to the right side

tan (a) + 3 = 0

Subtract

3

from both sides

tan (a) + 3 - 3 = 0 - 3

Simplify

tan (a) = - 3

tan (a) = - 3

General solutions for

tan (a) = - 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}

a = \frac{2 π}{3} + πn

a = \frac{2 π}{3} + πn

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

sin (a) - 3 cos (a) = 0

Rewrite using trig identities

sin (a) - 3 cos (a) = 0

Divide both sides by

cos (a), cos (a) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

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

tan (a) - 3 = 0

tan (a) - 3 = 0

Move

3

to the right side

tan (a) - 3 = 0

Add

3

to both sides

tan (a) - 3 + 3 = 0 + 3

Simplify

tan (a) = 3

tan (a) = 3

General solutions for

tan (a) = 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}

a = \frac{π}{3} + πn

a = \frac{π}{3} + πn

Combine all the solutions

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

Popular Examples

3cot(x)+sqrt(3)=0

3 cot (x) + 3 = 0

3cos(x)+9/2 =5+4cos(x)

3 cos (x) + \frac{9}{2} = 5 + 4 cos (x)

sec(2x)+sqrt(2)=0

sec (2 x) + 2 = 0

3sin(x+20)-0.75=0

3 sin (x + 2 0^{\circ}) - 0.75 = 0

2sin^2(x)+sin(x)-5=0

2 sin^{2} (x) + sin (x) - 5 = 0

Frequently Asked Questions (FAQ)

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