Question:
If the equation \(\cos^4\theta+\sin^4\theta+\lambda=0\) has real solutions, then \(\lambda\) lies in which interval?
If the equation \(\cos^4\theta+\sin^4\theta+\lambda=0\) has real solutions, then \(\lambda\) lies in which interval?
Show Hint
For expressions involving
\[
\sin^4\theta+\cos^4\theta,
\]
always convert to
\[
1-\frac12\sin^22\theta
\]
to obtain the range quickly.
Updated On: Jun 8, 2026
- \(\left(-\frac54,-1\right)\)
- \(\left[-1,-\frac12\right]\)
- \(\left(-\frac12,-\frac14\right]\)
- \(\left[-\frac32,-\frac54\right]\)
Show Solution
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The Correct Option is B
Solution and Explanation
Concept:
Use trigonometric identities to find the range of
\[
\cos^4\theta+\sin^4\theta.
\]
The equation will have real solutions only when \(\lambda\) cancels a value from this range.
Step 1: Simplify the expression.
\[ \cos^4\theta+\sin^4\theta = (\cos^2\theta+\sin^2\theta)^2 - 2\sin^2\theta\cos^2\theta \] \[ =1-2\sin^2\theta\cos^2\theta \] Using \[ \sin^2\theta\cos^2\theta = \frac14\sin^22\theta, \] we get \[ \cos^4\theta+\sin^4\theta = 1-\frac12\sin^22\theta \]
Step 2: Find the range.
Since \[ 0\le\sin^22\theta\le1, \] \[ \frac12 \le \cos^4\theta+\sin^4\theta \le 1 \]
Step 3: Use the equation.
Given \[ \cos^4\theta+\sin^4\theta+\lambda=0 \] \[ \lambda = -\left(\cos^4\theta+\sin^4\theta\right) \] Hence \[ -1 \le \lambda \le -\frac12 \] \[ \boxed{\lambda\in\left[-1,-\frac12\right]} \]
Step 1: Simplify the expression.
\[ \cos^4\theta+\sin^4\theta = (\cos^2\theta+\sin^2\theta)^2 - 2\sin^2\theta\cos^2\theta \] \[ =1-2\sin^2\theta\cos^2\theta \] Using \[ \sin^2\theta\cos^2\theta = \frac14\sin^22\theta, \] we get \[ \cos^4\theta+\sin^4\theta = 1-\frac12\sin^22\theta \]
Step 2: Find the range.
Since \[ 0\le\sin^22\theta\le1, \] \[ \frac12 \le \cos^4\theta+\sin^4\theta \le 1 \]
Step 3: Use the equation.
Given \[ \cos^4\theta+\sin^4\theta+\lambda=0 \] \[ \lambda = -\left(\cos^4\theta+\sin^4\theta\right) \] Hence \[ -1 \le \lambda \le -\frac12 \] \[ \boxed{\lambda\in\left[-1,-\frac12\right]} \]
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