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Popular Trigonometry >

4sin^2(θ)=3cos^2(θ),0<θ<360

Solution

4 sin^{2} (θ) = 3 cos^{2} (θ), 0^{\circ} < θ < 36 0^{\circ}

Solution

θ = - 0.71372 \dots + 18 0^{\circ}, θ = - 0.71372 \dots + 36 0^{\circ}, θ = 0.71372 \dots, θ = 0.71372 \dots + 18 0^{\circ}

Radians

θ = - 0.71372 \dots + π, θ = - 0.71372 \dots + 2 π, θ = 0.71372 \dots, θ = 0.71372 \dots + π

Solution steps

4 sin^{2} (θ) = 3 cos^{2} (θ), 0^{\circ} < θ < 36 0^{\circ}

Subtract

3 cos^{2} (θ)

from both sides

4 sin^{2} (θ) - 3 cos^{2} (θ) = 0

Factor

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

4 sin^{2} (θ) - 3 cos^{2} (θ)

Rewrite

4 sin^{2} (θ) - 3 cos^{2} (θ)

(2 sin (θ))^{2} - (3 cos (θ))^{2}

4 sin^{2} (θ) - 3 cos^{2} (θ)

Rewrite

4

2^{2}

= 2^{2} sin^{2} (θ) - 3 cos^{2} (θ)

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

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

Apply exponent rule:

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

2^{2} sin^{2} (θ) = (2 sin (θ))^{2}

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

Apply exponent rule:

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

(3)^{2} cos^{2} (θ) = (3 cos (θ))^{2}

= (2 sin (θ))^{2} - (3 cos (θ))^{2}

= (2 sin (θ))^{2} - (3 cos (θ))^{2}

Apply Difference of Two Squares Formula:

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

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

= (2 sin (θ) + 3 cos (θ)) (2 sin (θ) - 3 cos (θ))

(2 sin (θ) + 3 cos (θ)) (2 sin (θ) - 3 cos (θ)) = 0

Solving each part separately

2 sin (θ) + 3 cos (θ) = 0 or 2 sin (θ) - 3 cos (θ) = 0

2 sin (θ) + 3 cos (θ) = 0, 0 < θ < 36 0^{\circ} : θ = - arctan (\frac{3}{2}) + 18 0^{\circ}, θ = - arctan (\frac{3}{2}) + 36 0^{\circ}

2 sin (θ) + 3 cos (θ) = 0, 0 < θ < 36 0^{\circ}

Rewrite using trig identities

2 sin (θ) + 3 cos (θ) = 0

Divide both sides by

cos (θ), cos (θ) \neq = 0

\frac{2 sin ( θ ) + 3 cos ( θ )}{cos ( θ )} = \frac{0}{cos ( θ )}

Simplify

\frac{2 sin ( θ )}{cos ( θ )} + 3 = 0

Use the basic trigonometric identity:

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

2 tan (θ) + 3 = 0

2 tan (θ) + 3 = 0

Move

3

to the right side

2 tan (θ) + 3 = 0

Subtract

3

from both sides

2 tan (θ) + 3 - 3 = 0 - 3

Simplify

2 tan (θ) = - 3

2 tan (θ) = - 3

Divide both sides by

2

2 tan (θ) = - 3

Divide both sides by

2

\frac{2 tan ( θ )}{2} = \frac{- 3}{2}

Simplify

tan (θ) = - \frac{3}{2}

tan (θ) = - \frac{3}{2}

Apply trig inverse properties

tan (θ) = - \frac{3}{2}

General solutions for

tan (θ) = - \frac{3}{2}

tan (x) = - a \Rightarrow x = arctan (- a) + 18 0^{\circ} n

θ = arctan (- \frac{3}{2}) + 18 0^{\circ} n

θ = arctan (- \frac{3}{2}) + 18 0^{\circ} n

Solutions for the range

0 < θ < 36 0^{\circ}

θ = - arctan (\frac{3}{2}) + 18 0^{\circ}, θ = - arctan (\frac{3}{2}) + 36 0^{\circ}

2 sin (θ) - 3 cos (θ) = 0, 0 < θ < 36 0^{\circ} : θ = arctan (\frac{3}{2}), θ = arctan (\frac{3}{2}) + 18 0^{\circ}

2 sin (θ) - 3 cos (θ) = 0, 0 < θ < 36 0^{\circ}

Rewrite using trig identities

2 sin (θ) - 3 cos (θ) = 0

Divide both sides by

cos (θ), cos (θ) \neq = 0

\frac{2 sin ( θ ) - 3 cos ( θ )}{cos ( θ )} = \frac{0}{cos ( θ )}

Simplify

\frac{2 sin ( θ )}{cos ( θ )} - 3 = 0

Use the basic trigonometric identity:

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

2 tan (θ) - 3 = 0

2 tan (θ) - 3 = 0

Move

3

to the right side

2 tan (θ) - 3 = 0

Add

3

to both sides

2 tan (θ) - 3 + 3 = 0 + 3

Simplify

2 tan (θ) = 3

2 tan (θ) = 3

Divide both sides by

2

2 tan (θ) = 3

Divide both sides by

2

\frac{2 tan ( θ )}{2} = \frac{3}{2}

Simplify

tan (θ) = \frac{3}{2}

tan (θ) = \frac{3}{2}

Apply trig inverse properties

tan (θ) = \frac{3}{2}

General solutions for

tan (θ) = \frac{3}{2}

tan (x) = a \Rightarrow x = arctan (a) + 18 0^{\circ} n

θ = arctan (\frac{3}{2}) + 18 0^{\circ} n

θ = arctan (\frac{3}{2}) + 18 0^{\circ} n

Solutions for the range

0 < θ < 36 0^{\circ}

θ = arctan (\frac{3}{2}), θ = arctan (\frac{3}{2}) + 18 0^{\circ}

Combine all the solutions

θ = - arctan (\frac{3}{2}) + 18 0^{\circ}, θ = - arctan (\frac{3}{2}) + 36 0^{\circ}, θ = arctan (\frac{3}{2}), θ = arctan (\frac{3}{2}) + 18 0^{\circ}

Show solutions in decimal form

θ = - 0.71372 \dots + 18 0^{\circ}, θ = - 0.71372 \dots + 36 0^{\circ}, θ = 0.71372 \dots, θ = 0.71372 \dots + 18 0^{\circ}

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