What is the general solution for cos^2(a)-3sin^2(a)=0 ?

The general solution for cos^2(a)-3sin^2(a)=0 is a=(5pi)/6+pin,a= pi/6+pin

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Popular >

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

Solution

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

Solution

a = \frac{5 π}{6} + πn, a = \frac{π}{6} + πn

Degrees

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

Solution steps

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

Factor

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

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

Rewrite

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

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

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

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

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

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

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

Solving each part separately

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

cos (a) + 3 sin (a) = 0 : a = \frac{5 π}{6} + πn

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

1 + 3 tan (a) = 0

1 + 3 tan (a) = 0

Move

1

to the right side

1 + 3 tan (a) = 0

Subtract

1

from both sides

1 + 3 tan (a) - 1 = 0 - 1

Simplify

3 tan (a) = - 1

3 tan (a) = - 1

Divide both sides by

3

3 tan (a) = - 1

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 tan ( a )}{3} : tan (a)

\frac{3 tan ( a )}{3}

Cancel the common factor:

3

= tan (a)

Simplify

\frac{- 1}{3} : - \frac{3}{3}

\frac{- 1}{3}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1}{3}

Rationalize

- \frac{1}{3} : - \frac{3}{3}

- \frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= - \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{3}{3}

= - \frac{3}{3}

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

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

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

General solutions for

tan (a) = - \frac{3}{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{5 π}{6} + πn

a = \frac{5 π}{6} + πn

cos (a) - 3 sin (a) = 0 : a = \frac{π}{6} + πn

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

1 - 3 tan (a) = 0

1 - 3 tan (a) = 0

Move

1

to the right side

1 - 3 tan (a) = 0

Subtract

1

from both sides

1 - 3 tan (a) - 1 = 0 - 1

Simplify

- 3 tan (a) = - 1

- 3 tan (a) = - 1

Divide both sides by

- 3

- 3 tan (a) = - 1

Divide both sides by

- 3

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

Simplify

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

Simplify

\frac{- 3 tan ( a )}{- 3} : tan (a)

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

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

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

Cancel the common factor:

3

= tan (a)

Simplify

\frac{- 1}{- 3} : \frac{3}{3}

\frac{- 1}{- 3}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{1}{3}

Rationalize

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

= \frac{3}{3}

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

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

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

General solutions for

tan (a) = \frac{3}{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}{6} + πn

a = \frac{π}{6} + πn

Combine all the solutions

a = \frac{5 π}{6} + πn, a = \frac{π}{6} + πn

Graph

Popular Examples

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sin (α) = - \frac{3}{5}, π < α < \frac{3 π}{2}

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5+2tan(3θ+pi/3)=3,0<= θ<= pi

5 + 2 tan (3 θ + \frac{π}{3}) = 3, 0 \leq θ \leq π

sin(d)= 12/15

sin (d) = \frac{12}{15}

tan(θ^2)+tan(θ)=0

tan (θ^{2}) + tan (θ) = 0

Frequently Asked Questions (FAQ)

What is the general solution for cos^2(a)-3sin^2(a)=0 ?
The general solution for cos^2(a)-3sin^2(a)=0 is a=(5pi)/6+pin,a= pi/6+pin