Sure, let's solve the given expression step-by-step:
We need to evaluate [tex]\(\operatorname{Tan}\left(a+5^{\circ}\right) \)[/tex] given that:
[tex]\[
\operatorname{Tan}\left(a+5^{\circ}\right)=\sqrt{2 \sin 30^{\circ}+\sec^2 45^{\circ}}
\][/tex]
1. Calculate [tex]\(\sin 30^\circ\)[/tex]
[tex]\[
\sin 30^\circ = \frac{1}{2} = 0.5
\][/tex]
2. Calculate [tex]\(\sec 45^\circ\)[/tex]
The secant function is the reciprocal of the cosine function. Therefore:
[tex]\[
\sec 45^\circ = \frac{1}{\cos 45^\circ} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}
\][/tex]
3. Square [tex]\(\sec 45^\circ\)[/tex]
[tex]\[
\sec^2 45^\circ = (\sqrt{2})^2 = 2
\][/tex]
4. Substitute [tex]\(\sin 30^\circ\)[/tex] and [tex]\(\sec^2 45^\circ\)[/tex] into the expression inside the square root
[tex]\[
2 \sin 30^\circ + \sec^2 45^\circ = 2 \times 0.5 + 2 = 1 + 2 = 3
\][/tex]
5. Take the square root of the expression
[tex]\[
\sqrt{3} \approx 1.73205
\][/tex]
6. Evaluate [tex]\(\operatorname{Tan}\left(a+5^{\circ}\right)\)[/tex] where the value of [tex]\(a\)[/tex] is 5
We need to find the tan of the angle [tex]\(10^\circ\)[/tex] (since [tex]\(a + 5 = 5 + 5 = 10\)[/tex]):
[tex]\[
\tan(10^\circ) \approx 0.17632698070846498
\][/tex]
So, the step-by-step solution shows that:
[tex]\[
\operatorname{Tan}\left(a+5^{\circ}\right) \approx 0.17632698070846498
\][/tex]
Therefore:
[tex]\[
\operatorname{Tan}\left(10^\circ\right) = \sqrt{2 \sin 30^{\circ} + \sec^2 45^{\circ}} \approx 0.17632698070846498
\][/tex]
