`(sin70^(@)+cos40^(@))/(cos70^(@)+sin40^(@))=?`

Options

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

Answer

To solve the expression \((\sin 70^\circ + \cos 40^\circ) / (\cos 70^\circ + \sin 40^\circ)\), we can use the complementary angle identities and some trigonometric properties. Here’s a step-by-step solution:

### Step 1: Use complementary angle identities
We know that:
\[ \sin 70^\circ = \cos(90^\circ - 70^\circ) = \cos 20^\circ \]
\[ \cos 40^\circ = \sin(90^\circ - 40^\circ) = \sin 50^\circ \]
Thus, we can rewrite the expression as:
\[ \frac{\cos 20^\circ + \sin 50^\circ}{\cos 70^\circ + \sin 40^\circ} \]

### Step 2: Rewrite \(\cos 70^\circ\) and \(\sin 40^\circ\)
Similarly, we can express \(\cos 70^\circ\) and \(\sin 40^\circ\):
\[ \cos 70^\circ = \sin(90^\circ - 70^\circ) = \sin 20^\circ \]
\[ \sin 40^\circ = \cos(90^\circ - 40^\circ) = \cos 50^\circ \]
Now, substituting these into our expression gives:
\[ \frac{\cos 20^\circ + \sin 50^\circ}{\sin 20^\circ + \cos 50^\circ} \]

### Step 3: Simplify using angle sum identities
Now, we can use the sum-to-product identities:
\[ \cos 20^\circ + \sin 50^\circ = \cos 20^\circ + \cos(90^\circ - 50^\circ) = \cos 20^\circ + \cos 40^\circ \]
And for the denominator:
\[ \sin 20^\circ + \cos 50^\circ = \sin 20^\circ + \sin(90^\circ - 50^\circ) = \sin 20^\circ + \sin 40^\circ \]

### Step 4: Use the sine and cosine values
Now we can use known values:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 30^\circ = \frac{1}{2} \]
Thus, we can rewrite the expression as:
\[ \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \]

### Final Answer
The value of the expression \((\sin 70^\circ + \cos 40^\circ) / (\cos 70^\circ + \sin 40^\circ)\) is:
\[ \sqrt{3} \]

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