Trigonometry & Inverse Trigonometry
Trigonometric Identities
Grade 11
Question:
<p>If the equation \(\cos^4\theta + \sin^4\theta + \lambda = 0\) has real solutions for \(\theta\), then \(\lambda\) lies in the interval</p>
<p>(a) \(\left(-\frac{5}{4}, -1\right)\)</p>
<p>(b) \(\left[-1, -\frac{1}{2}\right]\)</p>
<p>(c) \(\left(-\frac{1}{2}, -\frac{1}{4}\right]\)</p>
<p>(d) \(\left[-\frac{3}{2}, -\frac{5}{4}\right]\)</p>
Step-by-Step Solution
Key Concept: Find the range of $\cos^4\theta + \sin^4\theta$ using algebraic identities, then determine the range of $\lambda$.
<p>Simplify $\cos^4\theta + \sin^4\theta = 1 - 2\sin^2\theta\cos^2\theta = 1 - \frac{1}{2}\sin^2 2\theta$. The range is $\left[\frac{1}{2}, 1\right]$. For the equation to have solutions, $\lambda = -(\cos^4\theta + \sin^4\theta)$ must lie in $[-1, -\frac{1}{2}]$.</p>
Correct Answer: B