To solve the given expression [tex]\(\frac{\sin 5^\circ}{\cos 35^\circ}+\frac{\cos 30^\circ}{\tan 60^\circ}\)[/tex], we'll break it down into two main components and evaluate them step by step.
Let's denote the expression as:
[tex]\[ E = \frac{\sin 5^\circ}{\cos 35^\circ} + \frac{\cos 30^\circ}{\tan 60^\circ} \][/tex]
### Step 1: Calculate [tex]\(\frac{\sin 5^\circ}{\cos 35^\circ}\)[/tex]
To begin, find the values of [tex]\(\sin 5^\circ\)[/tex] and [tex]\(\cos 35^\circ\)[/tex].
- For [tex]\(\sin 5^\circ\)[/tex]:
[tex]\[ \sin 5^\circ \approx 0.0872 \][/tex]
- For [tex]\(\cos 35^\circ\)[/tex]:
[tex]\[ \cos 35^\circ \approx 0.8192 \][/tex]
Next, compute the ratio:
[tex]\[ \frac{\sin 5^\circ}{\cos 35^\circ} = \frac{0.0872}{0.8192} \approx 0.1064 \][/tex]
### Step 2: Calculate [tex]\(\frac{\cos 30^\circ}{\tan 60^\circ}\)[/tex]
Now, evaluate the cosine and tangent components of the second part:
- For [tex]\(\cos 30^\circ\)[/tex]:
[tex]\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \approx 0.8660 \][/tex]
- For [tex]\(\tan 60^\circ\)[/tex]:
[tex]\[ \tan 60^\circ = \sqrt{3} \approx 1.7321 \][/tex]
Next, compute the ratio:
[tex]\[ \frac{\cos 30^\circ}{\tan 60^\circ} = \frac{0.8660}{1.7321} \approx 0.5000 \][/tex]
### Step 3: Combine the results
Finally, sum the two calculated values:
[tex]\[ \frac{\sin 5^\circ}{\cos 35^\circ} + \frac{\cos 30^\circ}{\tan 60^\circ} \approx 0.1064 + 0.5000 = 0.6064 \][/tex]
Hence, the value of the given expression [tex]\(\frac{\sin 5^\circ}{\cos 35^\circ}+\frac{\cos 30^\circ}{\tan 60^\circ}\)[/tex] is approximately [tex]\(0.6064\)[/tex].
