The value of `sin600^(@)cos330^(@)+cos120^(@)sin150^(@)` is

Options

A. 1
B. `-1`
C. `(1)/(sqrt(2))`
D. `(sqrt(3))/(2)`

Answer

To solve the expression \( \sin 60^\circ \cos 330^\circ + \cos 120^\circ \sin 150^\circ \), we will evaluate each trigonometric function step by step.

### Step 1: Evaluate \( \sin 60^\circ \)
The sine of \( 60^\circ \) is a well-known value:
\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \]

**Hint:** Remember the values of sine for common angles: \( 0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ \).

### Step 2: Evaluate \( \cos 330^\circ \)
To find \( \cos 330^\circ \), we can use the fact that \( 330^\circ \) is in the fourth quadrant:
\[ \cos 330^\circ = \cos(360^\circ - 30^\circ) = \cos 30^\circ = \frac{\sqrt{3}}{2} \]

**Hint:** In the fourth quadrant, cosine values are positive.

### Step 3: Evaluate \( \cos 120^\circ \)
The angle \( 120^\circ \) is in the second quadrant:
\[ \cos 120^\circ = \cos(180^\circ - 60^\circ) = -\cos 60^\circ = -\frac{1}{2} \]

**Hint:** In the second quadrant, cosine values are negative.

### Step 4: Evaluate \( \sin 150^\circ \)
The angle \( 150^\circ \) is also in the second quadrant:
\[ \sin 150^\circ = \sin(180^\circ - 30^\circ) = \sin 30^\circ = \frac{1}{2} \]

**Hint:** In the second quadrant, sine values are positive.

### Step 5: Substitute the values into the expression
Now we can substitute the values we found back into the original expression:
\[ \sin 60^\circ \cos 330^\circ + \cos 120^\circ \sin 150^\circ = \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(-\frac{1}{2}\right)\left(\frac{1}{2}\right) \]

### Step 6: Simplify the expression
Calculating each term:
\[ = \frac{3}{4} - \frac{1}{4} = \frac{3 - 1}{4} = \frac{2}{4} = \frac{1}{2} \]

### Final Answer
Thus, the value of \( \sin 60^\circ \cos 330^\circ + \cos 120^\circ \sin 150^\circ \) is:
\[ \frac{1}{2} \]

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