The polar form of sin `75^(@)+i cos 75^(@)` is

Options

A. (a) sin `75^(@)+icos 75^(@)`
B. (b) `sin 15^(@)+icos 15^(@)`
C. (c) `cos15^(@)+isin 15^(@)`
D. (d) `cos 75^(@)+isin75^(@)`

Answer

To find the polar form of \( \sin 75^\circ + i \cos 75^\circ \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ z = \sin 75^\circ + i \cos 75^\circ \]

### Step 2: Use co-function identities
We know from trigonometric identities that:
\[ \sin(90^\circ - \theta) = \cos \theta \quad \text{and} \quad \cos(90^\circ - \theta) = \sin \theta \]
Using these identities, we can rewrite \( \sin 75^\circ \) and \( \cos 75^\circ \):
\[ \sin 75^\circ = \cos(90^\circ - 75^\circ) = \cos 15^\circ \]
\[ \cos 75^\circ = \sin(90^\circ - 75^\circ) = \sin 15^\circ \]

### Step 3: Substitute back into the expression
Now substituting these values back into our expression for \( z \):
\[ z = \cos 15^\circ + i \sin 15^\circ \]

### Step 4: Identify the polar form
The expression \( \cos 15^\circ + i \sin 15^\circ \) is in the form of \( r(\cos \theta + i \sin \theta) \) where \( r = 1 \) and \( \theta = 15^\circ \). Therefore, the polar form is:
\[ z = 1 \cdot \left( \cos 15^\circ + i \sin 15^\circ \right) \]

### Conclusion
Thus, the polar form of \( \sin 75^\circ + i \cos 75^\circ \) is:
\[ \cos 15^\circ + i \sin 15^\circ \]

The correct answer is option **C: \( \cos 15^\circ + i \sin 15^\circ \)**.

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