Evaluate Trigonometric Expression with Angle Substitution

The problem asks us to find the value of the expression $\sin\mu + \cos(90^\circ - \mu)$ when the angle $\mu$ is given as $60^\circ$. We need to substitute the value of $\mu$ and simplify the expression using known trigonometric values and identities.

Step 1: Substitute the value of $\mu$

We are given $\mu = 60^\circ$. Substitute this value into the expression:

$$ \sin(60^\circ) + \cos(90^\circ - 60^\circ) $$

Step 2: Simplify the expression

First, simplify the angle inside the cosine function:

$$ 90^\circ - 60^\circ = 30^\circ $$

Now the expression becomes:

$$ \sin(60^\circ) + \cos(30^\circ) $$

Step 3: Evaluate the trigonometric functions

We know the standard trigonometric values for $60^\circ$ and $30^\circ$:

• $\sin(60^\circ) = \frac{\sqrt{3}}{2}$
• $\cos(30^\circ) = \frac{\sqrt{3}}{2}$

Alternatively, we could use the co-function identity $\cos(90^\circ - \mu) = \sin\mu$. In this case, $\cos(90^\circ - 60^\circ) = \sin(60^\circ) = \frac{\sqrt{3}}{2}$.

Step 4: Add the values

Now, add the values obtained in the previous step:

$$ \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} $$

To add these fractions, we keep the common denominator and add the numerators:

$$ \frac{\sqrt{3} + \sqrt{3}}{2} = \frac{2\sqrt{3}}{2} $$

Simplify the fraction by canceling the common factor of 2:

$$ \frac{2\sqrt{3}}{2} = \sqrt{3} $$

Step 5: Final Result

The value of the expression $\sin\mu + \cos(90^\circ - \mu)$ when $\mu = 60^\circ$ is $\sqrt{3}$.