Find `"sin" 7(1^(@))/(2),cos 7(1^(@))/(2)` and `"tan" 11(1^(@))/(4)`.

Answer

To solve the problem of finding \( \sin 7\frac{1}{2}^\circ \), \( \cos 7\frac{1}{2}^\circ \), and \( \tan 11\frac{1}{4}^\circ \), we will break it down step by step.

### Step 1: Convert Mixed Fractions to Improper Fractions
First, we convert the mixed fractions into improper fractions.

For \( 7\frac{1}{2}^\circ \):
\[ 7\frac{1}{2} = 7 + \frac{1}{2} = \frac{14}{2} + \frac{1}{2} = \frac{15}{2}^\circ \]

For \( 11\frac{1}{4}^\circ \):
\[ 11\frac{1}{4} = 11 + \frac{1}{4} = \frac{44}{4} + \frac{1}{4} = \frac{45}{4}^\circ \]

### Step 2: Finding \( \sin 7\frac{1}{2}^\circ \)
Now we need to find \( \sin \frac{15}{2}^\circ \).

Using the sine angle addition formula, we can express \( \sin \frac{15}{2}^\circ \) as:
\[ \sin \frac{15}{2}^\circ = \sin(7^\circ + \frac{1}{2}^\circ) \]

### Step 3: Applying the Sine Addition Formula
Using the sine addition formula:
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]
Let \( A = 7^\circ \) and \( B = \frac{1}{2}^\circ \):
\[ \sin \frac{15}{2}^\circ = \sin 7^\circ \cos \frac{1}{2}^\circ + \cos 7^\circ \sin \frac{1}{2}^\circ \]

### Step 4: Finding \( \cos 7\frac{1}{2}^\circ \)
Similarly, we can find \( \cos \frac{15}{2}^\circ \):
\[ \cos \frac{15}{2}^\circ = \cos(7^\circ + \frac{1}{2}^\circ) \]

Using the cosine addition formula:
\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]
So,
\[ \cos \frac{15}{2}^\circ = \cos 7^\circ \cos \frac{1}{2}^\circ - \sin 7^\circ \sin \frac{1}{2}^\circ \]

### Step 5: Finding \( \tan 11\frac{1}{4}^\circ \)
Now we find \( \tan \frac{45}{4}^\circ \):
\[ \tan \frac{45}{4}^\circ = \tan(11^\circ + \frac{1}{4}^\circ) \]
Using the tangent addition formula:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
Let \( A = 11^\circ \) and \( B = \frac{1}{4}^\circ \):
\[ \tan \frac{45}{4}^\circ = \frac{\tan 11^\circ + \tan \frac{1}{4}^\circ}{1 - \tan 11^\circ \tan \frac{1}{4}^\circ} \]

### Step 6: Final Values
Now we can substitute the known values of \( \sin 7^\circ \), \( \cos 7^\circ \), \( \sin \frac{1}{2}^\circ \), \( \cos \frac{1}{2}^\circ \), \( \tan 11^\circ \), and \( \tan \frac{1}{4}^\circ \) to get the final results for \( \sin 7\frac{1}{2}^\circ \), \( \cos 7\frac{1}{2}^\circ \), and \( \tan 11\frac{1}{4}^\circ \).