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Use an addition or subtraction formula to write the expression as a trigonometric function of one number.
[tex]\[ \sin \left(12^{\circ}\right) \cos \left(18^{\circ}\right) + \cos \left(12^{\circ}\right) \sin \left(18^{\circ}\right) \][/tex]

Find its exact value.
[tex]\[\square\][/tex]



Answer :

GinnyAnswer
Let's solve the given problem step by step.

### Step 1: Recognize the Trigonometric Identity
The given expression is:
[tex]\[ \sin(12^\circ) \cos(18^\circ) + \cos(12^\circ) \sin(18^\circ) \][/tex]

We can recognize that this expression matches the form of the sine addition formula:
[tex]\[ \sin(a) \cos(b) + \cos(a) \sin(b) = \sin(a + b) \][/tex]

Here, \( a = 12^\circ \) and \( b = 18^\circ \).

### Step 2: Apply the Sine Addition Formula
Using the sine addition formula, we can rewrite the expression as:
[tex]\[ \sin(12^\circ + 18^\circ) \][/tex]

### Step 3: Simplify the Angle
Now, compute the sum of the angles:
[tex]\[ 12^\circ + 18^\circ = 30^\circ \][/tex]

Therefore, the trigonometric function of one number is:
[tex]\[ \sin(30^\circ) \][/tex]

### Step 4: Find the Exact Value
We know from trigonometric values that:
[tex]\[ \sin(30^\circ) = \frac{1}{2} \][/tex]

So, the exact value of the expression is:
[tex]\[ \frac{1}{2} \][/tex]

### Summary
The given expression can be rewritten using the sine addition formula and simplified to:
[tex]\[ \sin(30^\circ) \][/tex]

The exact value of this expression is:
[tex]\[ \frac{1}{2} \][/tex]

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