Count Solutions of Trigonometric Equation in Interval

 

❓ Question

Find the number of solutions of the equation

$$\cos 2\theta \cos \frac{\theta }{2}+\cos \frac{5\theta }{2}=2{\cos }^3\frac{5\theta }{\mathrm{2}}$$

in the interval $[-\pi \mathrm{/}2,\pi \mathrm{/}2].$

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🖼️ Question Image

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✍️ Short Solution

Step 1 — Observe structure
The RHS has $2\cos^3\frac{5\theta}{2}$, which suggests the triple–angle identity:

$$\cos 3x=4{\cos }^{3}x-3\cos x\mathrm{.}$$

Step 2 — Rearrange equation
Move RHS to LHS:

$$\cos 2\theta \cos \frac{\theta }{2}+\cos \frac{5\theta }{2}-2{\cos }^3\frac{5\theta }{2}=0.$$

Write $2\cos^3 A = \tfrac12 (4\cos^3A) = \tfrac12(\cos3A + 3\cos A)$

So:

$$\cos 2\theta \cos \frac{\theta }{2}+\cos \frac{5\theta }{2}-\frac{1}{2}(\cos \frac{15\theta }{2}+3\cos \frac{5\theta }{2})=0.$$

Step 3 — Simplify LHS
After simplification, the equation reduces to:

$$\cos 2\theta \cos \frac{\theta }{2}+\cos \frac{5\theta }{2}-\frac{1}{2}\cos \frac{15\theta }{2}-\frac{3}{2}\cos \frac{5\theta }{2}=0.$$

Collect like terms:

$$\cos 2\theta \cos \frac{\theta }{2}-\frac{1}{2}\cos \frac{15\theta }{2}-\frac{1}{2}\cos \frac{5\theta }{2}=0.$$

Step 4 — Use product-to-sum

$$\cos 2\theta \cos \frac{\theta }{2}=\frac{1}{2}[\cos (2\theta -\frac{\theta }{2})+\cos (2\theta +\frac{\theta }{2})]=\frac{1}{2}[\cos \frac{3\theta }{2}+\cos \frac{5\theta }{2}]\mathrm{.}$$

So equation becomes:

$$\frac{1}{2}(\cos \frac{3\theta }{2}+\cos \frac{5\theta }{2})-\frac{1}{2}\cos \frac{15\theta }{2}-\frac{1}{2}\cos \frac{5\theta }{2}=0.$$

Multiply by 2:

$$\cos \frac{3\theta }{2}+\cos \frac{5\theta }{2}-\cos \frac{15\theta }{2}-\cos \frac{5\theta }{2}=0.$$

Simplify:

$$\cos \frac{3\theta }{2}-\cos \frac{15\theta }{2}=0.$$

Step 5 — Solve
$\cos A = \cos B$ gives $A = B + 2n\pi$ or $A = -B + 2n\pi$
Here $A = \frac{3\theta}{2},\; B = \frac{15\theta}{2}$.

1️⃣ $\frac{3\theta}{2} = \frac{15\theta}{2} + 2n\pi$
→ $-12\theta = 4n\pi$ → $\theta = -\frac{n\pi}{3}$

2️⃣ $\frac{3\theta}{2} = -\frac{15\theta}{2} + 2n\pi$
→ $9\theta = 2n\pi$ → $\theta = \frac{2n\pi}{9}$

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Step 6 — Count solutions in $[- \pi/2, \pi/2]$

• From case (1): $\theta = -\frac{n\pi}{3}$
  Values inside interval: $0,\; \pm\frac{\pi}{3}$(3 solutions)

• From case (2): $\theta = \frac{2n\pi}{9}$
  Range check: $-\frac{\pi}{2} \le \frac{2n\pi}{9} \le \frac{\pi}{2}$
  ⇒ $-\frac{9}{4} \le n \le \frac{9}{4}$ ⇒ $n=-2,-1,0,1,\mathrm{2(5\; solutions)}$

Total = $3+5=8$

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🖼️ Image Solution

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✅ Conclusion & Video Solution

The equation

$$\cos 2\theta \cos \frac{\theta }{2}+\cos \frac{5\theta }{2}=2{\cos }^3\frac{5\theta }{\mathrm{2}}$$

has 8 solutions in the interval $[-\pi/2,\,\pi/2]$

$$\boxed{{\displaystyle \text{Number of solutions}=\mathrm{8}}}$$

▶️ Watch full explanation video: