Question

Solve \(4 \sin ^{2} x=3 \tan ^{2} x-1\) algebraically. Give the general solution expressed in radians.

Short answer

The general solution is \( x = \frac{\text{pi}}{4} + 2n\text{pi}, \frac{3\text{pi}}{4} + 2n\text{pi}, -\frac{\text{pi}}{4} + 2n\text{pi}, \frac{5\text{pi}}{4} + 2n\text{pi} \).

Step-by-step solution

Use trigonometric identities

Rewrite the equation using the identity \(\tan x = \frac{\text{sin} x}{\text{cos} x}\) and \( \tan^2 x = \frac{\text{sin}^2 x}{\text{cos}^2 x} \). Substitute \(\frac{\text{sin}^2 x}{\text{cos}^2 x}\) for \( \tan^2 x \). The equation becomes: \[ 4 \text{sin}^2 x = 3 \frac{\text{sin}^2 x}{\text{cos}^2 x} - 1 \]

Clear the fraction

Multiply every term by \( \text{cos}^2 x \) to eliminate the fraction: \[ 4 \text{sin}^2 x \text{cos}^2 x = 3 \text{sin}^2 x - \text{cos}^2 x \]

Use Pythagorean identity

Apply the Pythagorean identity \( \text{sin}^2 x + \text{cos}^2 x = 1 \) to simplify the equation. \[ 4 \text{sin}^2 x (1 - \text{sin}^2 x) = 3 \text{sin}^2 x - ( 1 - \text{sin}^2 x ) \]

Simplify

Distribute and combine like terms: \[ 4 \text{sin}^2 x - 4 \text{sin}^4 x = 3 \text{sin}^2 x - 1 + \text{sin}^2 x \] Simplify further to: \[ 4 \text{sin}^2 x - 4 \text{sin}^4 x = 4 \text{sin}^2 x - 1 \]

Solve for \( \text{sin}^2 x \)

Isolate the \( \text{sin}^2 x \) terms on one side: \[ -4 \text{sin}^4 x = -1 \] Divide both sides by -4: \[ \text{sin}^4 x = \frac{1}{4} \] Take the square root of both sides: \[ \text{sin}^2 x = \frac{1}{2} \]

Solve for \( \text{sin} x \)

Take the square root of both sides again to solve for \( \text{sin} x \): \[ \text{sin} x = \frac{\text{sqrt}(2)}{2} \] or \[ \text{sin} x = -\frac{\text{sqrt}(2)}{2} \]

Find general solutions

The general solutions for \( \text{sin} x = \frac{\text{sqrt}(2)}{2} \) are \( x = \frac{\text{pi}}{4} + 2n\text{pi} \text{ or } x = \frac{3\text{pi}}{4} + 2n\text{pi} \). The general solutions for \( \text{sin} x = -\frac{\text{sqrt}(2)}{2} \) are \( x = -\frac{\text{pi}}{4} + 2n\text{pi} \text{ or } x = \frac{5\text{pi}}{4} + 2n\text{pi} \), where \( n \) is an integer.

Key concepts

trigonometric identities

Trigonometric identities are essential in simplifying and solving trigonometric equations. They express relationships between different trigonometric functions and can be very helpful in transforming complex expressions into more manageable forms.
For example, the identity for tangent, \(\tan x = \frac{\text{sin} x}{\text{cos} x}\), can convert problems involving tangent into problems involving sine and cosine, which might be easier to handle.

When you square both sides of \(\tan x = \frac{\text{sin} x}{\text{cos} x}\), you get \(\tan^2 x = \frac{\text{sin}^2 x}{\text{cos}^2 x}\). This identity allows us to rewrite expressions involving \(\tan^2 x\) in terms of \(\text{sin}^2 x\) and \(\text{cos}^2 x\).
This comes in handy, especially when combined with other identities
such as the Pythagorean identity.

Pythagorean identity

The Pythagorean identity is a critical tool in trigonometry that states \(\text{sin}^2 x + \text{cos}^2 x = 1\).
This identity allows us to express one trigonometric function in terms of another.
It's derived from the Pythagorean theorem and holds for all angles.

In our exercise, the Pythagorean identity is used to eliminate \(\text{cos}^2 x\) from the equation.
By replacing \(\text{cos}^2 x\) with \((1 - \text{sin}^2 x)\),
we convert everything into \(\text{sin}^2 x\), simplifying the equation to a point where we can solve for \(\text{sin}^2 x\).
The identity helps us move from a complex equation to a simpler quadratic equation.
From there, we can isolate \(\text{sin}^2 x\) and solve for \(\text{sin} x\).

general solutions

When solving trigonometric equations, we often seek general solutions.
This means finding all possible angles that satisfy the equation, not just specific ones.

For the equation \(4 \text{sin}^2 x=3 \tan^2 x-1\),
we solve for \( \text{sin} x\) and then determine all angles x that satisfy \( \text{sin} x= \frac{ \text{sqrt}(2)}{2}\) or \( \text{sin} x = - \frac{ \text{sqrt}(2)}{2}\).

The general solutions for these values of sine are:
\( x = \frac{\text{pi}}{4} + 2n\text{pi}\) or \( x = \frac{3\text{pi}}{4} + 2n\text{pi}\)
for \( \text{sin} x = \frac{\text{sqrt}(2)}{2}\),
and \( x = -\frac{\text{pi}}{4} + 2n\text{pi}\) or \( x = \frac{5\text{pi}}{4} + 2n\text{pi}\)
for \( \text{sin} x = -\frac{\text{sqrt}(2)}{2}\).

The variable \( n\) represents any integer, encapsulating all possible solutions in the form of a general formula.