JAMIA 2026 — Mathematics PYQ

Method 2: Algebraic Derivation

If you want to solve it step-by-step using identities:

1. Rearrange the equation:

$$3\sin \theta =5-4\cos \theta$$

2. Square both sides:

$$(3\sin \theta {)}^2=(5-4\cos \theta {)}^2$$

$$9{\sin }^2\theta =25-40\cos \theta +16{\cos }^2\theta$$

3. Substitute ${\sin }^2\theta =1-{\cos }^2\theta$:

$$9(1-{\cos }^2\theta )=25-40\cos \theta +16{\cos }^2\theta$$

$$9-9{\cos }^2\theta =25-40\cos \theta +16{\cos }^2\theta$$

4. Form a quadratic equation in terms of $\cos \theta$:

$$25{\cos }^2\theta -40\cos \theta +16=0$$

$$(5\cos \theta -4{)}^2=0$$

$$5\cos \theta -4=0⟹\cos \theta =\frac{4}{5}$$

5. Find $\sin \theta$:

$$\sin \theta =\sqrt{1-{\cos }^2\theta }=\sqrt{1-{\left(\frac{4}{5}\right)}^2}=\sqrt{1-\frac{16}{25}}=\sqrt{\frac{9}{25}}=\frac{3}{5}$$

Correct Answer

Option (b) 3/5