What is the general solution for 2tan(θ)cos^2(θ)=1 ?

The general solution for 2tan(θ)cos^2(θ)=1 is θ= pi/4+pin

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Popular >

2tan(θ)cos^2(θ)=1

Solution

2 tan (θ) cos^{2} (θ) = 1

Solution

θ = \frac{π}{4} + πn

Degrees

θ = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

2 tan (θ) cos^{2} (θ) = 1

Subtract

1

from both sides

2 cos^{2} (θ) tan (θ) - 1 = 0

Rewrite using trig identities

- 1 + 2 cos^{2} (θ) tan (θ)

Use the basic trigonometric identity:

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

= - 1 + 2 cos^{2} (θ) \frac{sin ( θ )}{cos ( θ )}

2 cos^{2} (θ) \frac{sin ( θ )}{cos ( θ )} = 2 sin (θ) cos (θ)

2 cos^{2} (θ) \frac{sin ( θ )}{cos ( θ )}

Multiply fractions:

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

= \frac{sin ( θ ) \cdot 2 cos ^{2} ( θ )}{cos ( θ )}

Cancel the common factor:

cos (θ)

= 2 sin (θ) cos (θ)

= - 1 + 2 sin (θ) cos (θ)

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

= - 1 + sin (2 θ)

- 1 + sin (2 θ) = 0

Move

1

to the right side

- 1 + sin (2 θ) = 0

Add

1

to both sides

- 1 + sin (2 θ) + 1 = 0 + 1

Simplify

sin (2 θ) = 1

sin (2 θ) = 1

General solutions for

sin (2 θ) = 1

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

2 θ = \frac{π}{2} + 2 πn

2 θ = \frac{π}{2} + 2 πn

Solve

2 θ = \frac{π}{2} + 2 πn : θ = \frac{π}{4} + πn

2 θ = \frac{π}{2} + 2 πn

Divide both sides by

2

2 θ = \frac{π}{2} + 2 πn

Divide both sides by

2

\frac{2 θ}{2} = \frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 θ}{2} = \frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2} : \frac{π}{4} + πn

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{2}}{2} = \frac{π}{4}

\frac{\frac{π}{2}}{2}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π}{4}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{4} + πn

θ = \frac{π}{4} + πn

θ = \frac{π}{4} + πn

θ = \frac{π}{4} + πn

θ = \frac{π}{4} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 2tan(θ)cos^2(θ)=1 ?
The general solution for 2tan(θ)cos^2(θ)=1 is θ= pi/4+pin