What is the general solution for 9(cos(a))^2-9(sin(a))^2=0 ?

The general solution for 9(cos(a))^2-9(sin(a))^2=0 is a=(3pi)/4+pin,a= pi/4+pin

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9(cos(a))^2-9(sin(a))^2=0

Solution

9 (cos (a))^{2} - 9 (sin (a))^{2} = 0

Solution

a = \frac{3 π}{4} + πn, a = \frac{π}{4} + πn

Degrees

a = 13 5^{\circ} + 18 0^{\circ} n, a = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

9 (cos (a))^{2} - 9 (sin (a))^{2} = 0

Factor

9 (cos (a))^{2} - 9 (sin (a))^{2} : 9 (cos (a) + sin (a)) (cos (a) - sin (a))

9 cos^{2} (a) - 9 sin^{2} (a)

Factor out common term

9

= 9 (cos^{2} (a) - sin^{2} (a))

Factor

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

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

Apply Difference of Two Squares Formula:

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

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

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

= 9 (cos (a) + sin (a)) (cos (a) - sin (a))

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

Solving each part separately

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

cos (a) + sin (a) = 0 : a = \frac{3 π}{4} + πn

cos (a) + sin (a) = 0

Rewrite using trig identities

cos (a) + sin (a) = 0

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

1 + tan (a) = 0

1 + tan (a) = 0

Move

1

to the right side

1 + tan (a) = 0

Subtract

1

from both sides

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

Simplify

tan (a) = - 1

tan (a) = - 1

General solutions for

tan (a) = - 1

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{3 π}{4} + πn

a = \frac{3 π}{4} + πn

cos (a) - sin (a) = 0 : a = \frac{π}{4} + πn

cos (a) - sin (a) = 0

Rewrite using trig identities

cos (a) - sin (a) = 0

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

1 - tan (a) = 0

1 - tan (a) = 0

Move

1

to the right side

1 - tan (a) = 0

Subtract

1

from both sides

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

Simplify

- tan (a) = - 1

- tan (a) = - 1

Divide both sides by

- 1

- tan (a) = - 1

Divide both sides by

- 1

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

Simplify

tan (a) = 1

tan (a) = 1

General solutions for

tan (a) = 1

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}{4} + πn

a = \frac{π}{4} + πn

Combine all the solutions

a = \frac{3 π}{4} + πn, a = \frac{π}{4} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 9(cos(a))^2-9(sin(a))^2=0 ?
The general solution for 9(cos(a))^2-9(sin(a))^2=0 is a=(3pi)/4+pin,a= pi/4+pin