$ ext{If } \alpha, \beta, \gamma ext{ are acute angles such that } \sin \alpha = \frac{\sqrt{3}}{2}, \cos \beta = \frac{\sqrt{3}}{2}, ext{ and } an \gamma = 1, ext{then what is } \alpha + \beta + \gamma ext{ equal to :}$

Question

$	ext{If } \alpha, \beta, \gamma 	ext{ are acute angles such that } \sin \alpha = \frac{\sqrt{3}}{2}, \cos \beta = \frac{\sqrt{3}}{2}, 	ext{ and } 	an \gamma = 1, 	ext{then what is } \alpha + \beta + \gamma 	ext{ equal to :}$

Accepted Answer

To determine why Option 3 ($135^\circ$) is the correct answer for the given problem, let's analyze the information provided and calculate the values of $\alpha$, $\beta$, and $\gamma$.

1. **Finding $\alpha$:**
   - We are given that $\sin \alpha = \frac{\sqrt{3}}{2}$.
   - The sine of an angle is $\frac{\sqrt{3}}{2}$ at $60^\circ$ in the first quadrant (since $\alpha$ is an acute angle).
   - Therefore, $\alpha = 60^\circ$.

2. **Finding $\beta$:**
   - We are given that $\cos \beta = \frac{\sqrt{3}}{2}$.
   - The cosine of an angle is $\frac{\sqrt{3}}{2}$ at $30^\circ$ in the first quadrant (since $\beta$ is an acute angle).
   - Therefore, $\beta = 30^\circ$.

3. **Finding $\gamma$:**
   - We are given that $\tan \gamma = 1$.
   - The tangent of an angle is $1$ at $45^\circ$ in the first quadrant (since $\gamma$ is an acute angle).
   - Therefore, $\gamma = 45^\circ$.

Now, we can calculate $\alpha + \beta + \gamma$:

$$
\alpha + \beta + \gamma = 60^\circ + 30^\circ + 45^\circ = 135^\circ
$$

Thus, the sum of the angles $\alpha$, $\beta$, and $\gamma$ is $135^\circ$, which corresponds to Option 3.

**Explanation for why other options are incorrect:**

- **Option 1 ($105^\circ$):** This option is incorrect because the sum of $60^\circ$, $30^\circ$, and $45^\circ$ is not $105^\circ$.
- **Option 2 ($120^\circ$):** This option is incorrect because the sum of $60^\circ$, $30^\circ$, and $45^\circ$ is not $120^\circ$.
- **Option 4 ($150^\circ$):** This option is incorrect because the sum of $60^\circ$, $30^\circ$, and $45^\circ$ is not $150^\circ$.

Therefore, based on the calculations and logical reasoning, Option 3 ($135^\circ$) is indeed the correct answer.