Understanding the Trigonometry Problem

The question asks us to evaluate a trigonometric expression involving sine and cosine functions with specific angle values. We are given that \(\alpha = 15^\circ\) and \(\beta = 15^\circ\). We need to find the value of \(\sin (\alpha + 1^\circ) + \cos (\beta + 1^\circ)\).

Since \(\alpha = 15^\circ\) and \(\beta = 15^\circ\), the expression becomes:

Our goal is to simplify this sum and express it in one of the forms provided in the options. The options involve terms like \(\cos 1^\circ\) and \(\sin 1^\circ\), which suggests using angle addition formulas based on the \(15^\circ\) and \(1^\circ\) components of \(16^\circ\).

Applying Angle Addition Formulas

We will use the angle addition formulas for sine and cosine:

• \(\sin(A + B) = \sin A \cos B + \cos A \sin B\)
• \(\cos(A + B) = \cos A \cos B - \sin A \sin B\)

Applying these to \(\sin(16^\circ)\) and \(\cos(16^\circ)\) with \(A = 15^\circ\) and \(B = 1^\circ\):

Now, we add these two expressions:

Let's group the terms with \(\cos 1^\circ\) and \(\sin 1^\circ\):

To proceed, we need the values of \(\sin 15^\circ\) and \(\cos 15^\circ\).

Calculating sin 15° and cos 15°

We can calculate these values using the angles \(45^\circ\) and \(30^\circ\), since \(15^\circ = 45^\circ - 30^\circ\). Using the angle subtraction formulas:

• \(\sin(A - B) = \sin A \cos B - \cos A \sin B\)
• \(\cos(A - B) = \cos A \cos B + \sin A \sin B\)

For \(A = 45^\circ\) and \(B = 30^\circ\):

Substitute the known values:

Similarly for \(\cos 15^\circ\):

Substitute the known values:

Substituting and Simplifying the Expression

Now, substitute the values of \(\sin 15^\circ\) and \(\cos 15^\circ\) back into the expression for \(\sin(16^\circ) + \cos(16^\circ)\):

Simplify the terms in the parentheses:

Finally, factor out \(\frac{1}{\sqrt{2}}\) to match the format of the options:

This result matches option 4.

Comparing with Options

Let's compare our derived expression with the given options:

• Option 1: \(\sqrt{3} \cos 1^\circ + \sin 1^\circ\)
• Option 2: \(\sqrt{3} \cos 1^\circ - \frac{1}{2} \sin 1^\circ\)
• Option 3: \(\frac{1}{\sqrt{2}} (\sqrt{3} \cos 1^\circ - \sin 1^\circ)\)
• Option 4: \(\frac{1}{\sqrt{2}} (\sqrt{3} \cos 1^\circ + \sin 1^\circ)\)

Our result \(\frac{1}{\sqrt{2}} (\sqrt{3} \cos 1^\circ + \sin 1^\circ)\) is identical to Option 4.

Revision Table: Key Trigonometry Concepts

Additional Information on Trigonometric Expressions

Expressions of the form \(a \sin x + b \cos x\) can always be rewritten in the form \(R \sin(x + \phi)\) or \(R \cos(x - \phi)\), where \(R = \sqrt{a^2 + b^2}\). This transformation is useful for finding maximum/minimum values or simplifying expressions.

In our case, for \(\sin(16^\circ) + \cos(16^\circ)\), \(a=1\) and \(b=1\). So, \(R = \sqrt{1^2 + 1^2} = \sqrt{2}\). The expression can be written as \(\sqrt{2} \left(\frac{1}{\sqrt{2}} \sin 16^\circ + \frac{1}{\sqrt{2}} \cos 16^\circ\right)\). Recognizing \(\frac{1}{\sqrt{2}} = \cos 45^\circ = \sin 45^\circ\), we get \(\sqrt{2} (\cos 45^\circ \sin 16^\circ + \sin 45^\circ \cos 16^\circ) = \sqrt{2} \sin(16^\circ + 45^\circ) = \sqrt{2} \sin(61^\circ)\). This is an alternative form of the answer, but it does not match the structure of the given options which explicitly contain \(\cos 1^\circ\) and \(\sin 1^\circ\).

Therefore, the approach using angle addition formulas on \(16^\circ = 15^\circ + 1^\circ\) was necessary to arrive at the form presented in the options.