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

The general solution for 3tan^2(x)-cos^2(x)=sin^2(x) is x=0.52359…+pin,x=-0.52359…+pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

3tan^2(x)-cos^2(x)=sin^2(x)

Solution

3 tan^{2} (x) - cos^{2} (x) = sin^{2} (x)

Solution

x = 0.52359 \dots + πn, x = - 0.52359 \dots + πn

Degrees

x = 3 0^{\circ} + 18 0^{\circ} n, x = - 3 0^{\circ} + 18 0^{\circ} n

Solution steps

3 tan^{2} (x) - cos^{2} (x) = sin^{2} (x)

Subtract

sin^{2} (x)

from both sides

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

Rewrite using trig identities

- cos^{2} (x) - sin^{2} (x) + 3 tan^{2} (x)

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

- cos^{2} (x) - sin^{2} (x) = - 1

= 3 tan^{2} (x) - 1

- 1 + 3 tan^{2} (x) = 0

Solve by substitution

- 1 + 3 tan^{2} (x) = 0

Let:

tan (x) = u

- 1 + 3 u^{2} = 0

- 1 + 3 u^{2} = 0 : u = \frac{1}{3}, u = - \frac{1}{3}

- 1 + 3 u^{2} = 0

Move

1

to the right side

- 1 + 3 u^{2} = 0

Add

1

to both sides

- 1 + 3 u^{2} + 1 = 0 + 1

Simplify

3 u^{2} = 1

3 u^{2} = 1

Divide both sides by

3

3 u^{2} = 1

Divide both sides by

3

\frac{3 u ^{2}}{3} = \frac{1}{3}

Simplify

u^{2} = \frac{1}{3}

u^{2} = \frac{1}{3}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{1}{3}, u = - \frac{1}{3}

Substitute back

u = tan (x)

tan (x) = \frac{1}{3}, tan (x) = - \frac{1}{3}

tan (x) = \frac{1}{3}, tan (x) = - \frac{1}{3}

tan (x) = \frac{1}{3} : x = arctan (\frac{1}{3}) + πn

tan (x) = \frac{1}{3}

Apply trig inverse properties

tan (x) = \frac{1}{3}

General solutions for

tan (x) = \frac{1}{3}

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{1}{3}) + πn

x = arctan (\frac{1}{3}) + πn

tan (x) = - \frac{1}{3} : x = arctan (- \frac{1}{3}) + πn

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

Apply trig inverse properties

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

General solutions for

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

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- \frac{1}{3}) + πn

x = arctan (- \frac{1}{3}) + πn

Combine all the solutions

x = arctan (\frac{1}{3}) + πn, x = arctan (- \frac{1}{3}) + πn

Show solutions in decimal form

x = 0.52359 \dots + πn, x = - 0.52359 \dots + πn

Popular Examples

cos(θ)=(2pi)/3

cos (θ) = \frac{2 π}{3}

3(tan(θ)-2)=2tan(-7)

3 (tan (θ) - 2) = 2 tan (- 7)

4cos^2(x)+7cos(x)=2

4 cos^{2} (x) + 7 cos (x) = 2

sin(x)-2cos(x)sin(x)=0

sin (x) - 2 cos (x) sin (x) = 0

3/5 =cos(x)

\frac{3}{5} = cos (x)

Frequently Asked Questions (FAQ)

What is the general solution for 3tan^2(x)-cos^2(x)=sin^2(x) ?
The general solution for 3tan^2(x)-cos^2(x)=sin^2(x) is x=0.52359…+pin,x=-0.52359…+pin