Find the value of θ (1° ≤ θ ≤ 90°) that satisfies the equation √3 cos θ + sin θ = 1.

Question

Accepted Answer

We are given:

$$
\sqrt{3} \cos 	heta + \sin 	heta = 1
$$

This is a linear combination of sine and cosine.

Let’s write it in the standard form:

$$
a \cos 	heta + b \sin 	heta = c
$$

Here:

* $a = \sqrt{3}$
* $b = 1$
* $c = 1$

---

We use the identity:

$$
a \cos 	heta + b \sin 	heta = R \cos(	heta - \alpha)
$$

Where:

* $R = \sqrt{a^2 + b^2} = \sqrt{3 + 1} = 2$
* $\cos \alpha = \frac{a}{R} = \frac{\sqrt{3}}{2}$
* $\sin \alpha = \frac{b}{R} = \frac{1}{2}$

So,

$$
\sqrt{3} \cos 	heta + \sin 	heta = 2 \cos(	heta - 30^\circ)
$$

Now set it equal to 1:

$$
2 \cos(	heta - 30^\circ) = 1
\Rightarrow \cos(	heta - 30^\circ) = \frac{1}{2}
$$

This gives:

$$
	heta - 30^\circ = 60^\circ \Rightarrow 	heta = 90^\circ
$$

Also,

$$
	heta - 30^\circ = 300^\circ \Rightarrow 	heta = 330^\circ 	ext{ (not valid as } 	heta \leq 90^\circ)
$$

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### Final Answer:

$$
\boxed{90^\circ}
$$

Answer

Answer

Answer