If `sin(theta+23^(@))=cos58^(@)`, then what is the value of cos `5theta` ?

Options

A. `(1)/(2)`
B. `(1)/(sqrt(2))`
C. `(sqrt(3))/(2)`
D. 0

Answer

To solve the equation \( \sin(\theta + 23^\circ) = \cos(58^\circ) \), we can use the co-function identity of sine and cosine, which states that:

\[ \sin(x) = \cos(90^\circ - x) \]

### Step 1: Rewrite the equation using the co-function identity
Using the identity, we can rewrite the equation as:

\[ \sin(\theta + 23^\circ) = \cos(58^\circ) \implies \sin(\theta + 23^\circ) = \sin(90^\circ - 58^\circ) \]

### Step 2: Simplify the right side
Calculate \( 90^\circ - 58^\circ \):

\[ 90^\circ - 58^\circ = 32^\circ \]

So, we have:

\[ \sin(\theta + 23^\circ) = \sin(32^\circ) \]

### Step 3: Set up the equations
Since the sine function is periodic, we can set up the following equations:

1. \( \theta + 23^\circ = 32^\circ + n \cdot 360^\circ \) (for any integer \( n \))
2. \( \theta + 23^\circ = 180^\circ - 32^\circ + n \cdot 360^\circ \)

### Step 4: Solve the first equation
From the first equation:

\[ \theta + 23^\circ = 32^\circ \implies \theta = 32^\circ - 23^\circ = 9^\circ \]

### Step 5: Solve the second equation
From the second equation:

\[ \theta + 23^\circ = 180^\circ - 32^\circ \implies \theta + 23^\circ = 148^\circ \implies \theta = 148^\circ - 23^\circ = 125^\circ \]

### Step 6: Determine \( \cos(5\theta) \)
Now, we need to find \( \cos(5\theta) \). We will calculate it for both values of \( \theta \).

1. For \( \theta = 9^\circ \):

\[ 5\theta = 5 \times 9^\circ = 45^\circ \implies \cos(5\theta) = \cos(45^\circ) = \frac{1}{\sqrt{2}} \]

2. For \( \theta = 125^\circ \):

\[ 5\theta = 5 \times 125^\circ = 625^\circ \]

Now, reduce \( 625^\circ \) to an equivalent angle within \( 0^\circ \) to \( 360^\circ \):

\[ 625^\circ - 360^\circ = 265^\circ \implies \cos(625^\circ) = \cos(265^\circ) \]

Since \( \cos(265^\circ) = -\sin(5^\circ) \), and \( \cos(5^\circ) \) is positive, we find that:

\[ \cos(265^\circ) \text{ is negative, and thus does not yield a valid answer.} \]

### Final Answer
Thus, the value of \( \cos(5\theta) \) is:

\[ \cos(5\theta) = \frac{1}{\sqrt{2}} \]