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

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

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Popular Trigonometry >

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

Solution

3 \frac{cos ^{2} ( a )}{sin ^{2} ( a )} = (cot (a))^{2}

Solution

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

Degrees

a = 9 0^{\circ} + 36 0^{\circ} n, a = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

3 \cdot \frac{cos ^{2} ( a )}{sin ^{2} ( a )} = (cot (a))^{2}

Subtract

(cot (a))^{2}

from both sides

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

Simplify

\frac{3 cos ^{2} ( a )}{sin ^{2} ( a )} - cot^{2} (a) : \frac{3 cos ^{2} ( a ) - cot ^{2} ( a ) sin ^{2} ( a )}{sin ^{2} ( a )}

\frac{3 cos ^{2} ( a )}{sin ^{2} ( a )} - cot^{2} (a)

Convert element to fraction:

cot^{2} (a) = \frac{cot ^{2} ( a ) sin ^{2} ( a )}{sin ^{2} ( a )}

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

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Factor

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

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

Rewrite

cot^{2} (a) sin^{2} (a)

(cot (a) sin (a))^{2}

cot^{2} (a) sin^{2} (a)

Apply exponent rule:

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

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

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

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

Rewrite

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

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

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

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

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

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

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

Solving each part separately

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

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

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

Rewrite using trig identities

cos (a) 3 + cot (a) sin (a)

Use the basic trigonometric identity:

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

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

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

\frac{cos ( a )}{sin ( a )} sin (a)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Cancel the common factor:

sin (a)

= cos (a)

= 3 cos (a) + cos (a)

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

Solve by substitution

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

Let:

cos (a) = u

u + u 3 = 0

u + u 3 = 0 : u = 0

u + u 3 = 0

Factor

u + u 3 : (1 + 3) u

u + u 3

Factor out common term

u

= u (1 + 3)

(1 + 3) u = 0

Divide both sides by

1 + 3

(1 + 3) u = 0

Divide both sides by

1 + 3

\frac{( 1 + 3 ) u}{1 + 3} = \frac{0}{1 + 3}

Simplify

u = 0

u = 0

Substitute back

u = cos (a)

cos (a) = 0

cos (a) = 0

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

cos (a) = 0

General solutions for

cos (a) = 0

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}

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

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

Combine all the solutions

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

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

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

Rewrite using trig identities

cos (a) 3 - cot (a) sin (a)

Use the basic trigonometric identity:

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

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

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

\frac{cos ( a )}{sin ( a )} sin (a)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Cancel the common factor:

sin (a)

= cos (a)

= 3 cos (a) - cos (a)

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

Solve by substitution

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

Let:

cos (a) = u

- u + u 3 = 0

- u + u 3 = 0 : u = 0

- u + u 3 = 0

Factor

- u + u 3 : (- 1 + 3) u

- u + u 3

Factor out common term

u

= u (- 1 + 3)

(- 1 + 3) u = 0

Divide both sides by

- 1 + 3

(- 1 + 3) u = 0

Divide both sides by

- 1 + 3

\frac{( - 1 + 3 ) u}{- 1 + 3} = \frac{0}{- 1 + 3}

Simplify

u = 0

u = 0

Substitute back

u = cos (a)

cos (a) = 0

cos (a) = 0

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

cos (a) = 0

General solutions for

cos (a) = 0

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}

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

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

Combine all the solutions

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

Combine all the solutions

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

Graph

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Frequently Asked Questions (FAQ)

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