Let's start with tangent!

Question:
Calculate $(\tan 15^o + \sqrt{3})^2$ without a calculator.

Answer:
It is not easy to obtain $\tan 15^o$ without a calculator. In this case, you may want to come up with angles which can easily convert into specific numbers, such as 30, 45, 60, 90, etc. In fact, 15 = 45 - 30, so $\tan 15^o = \tan(45^o - 30^o)$. Thus, we can use the addition theorem.
\[ \tan(45^o - 30^o) = \frac{\tan 45^o - \tan 30^o}{1 + \tan 45^o \tan 30^o} \]
If we use typical triangles, we know $\tan 45^o = 1$ and $\tan 30^o = 1/\sqrt{3}$. Therefore,
\[ \tan 15^o = \frac{1 - 1/\sqrt{3}}{1 + 1 \times 1/\sqrt{3}} \]
Multiply $\sqrt{3}$ by both numerator and denominator.
\[ = \frac{1 - 1/\sqrt{3}}{1 + 1/\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} \]
\[ = \frac{\sqrt{3}-1}{\sqrt{3}+1} \]
\[ = \frac{\sqrt{3}-1}{\sqrt{3}+1} \times \frac{\sqrt{3}-1}{\sqrt{3}-1} \]
\[ = \frac{4 - 2\sqrt{3}}{2} = 2 - \sqrt{3}\]
Hence, we have
\[ (\tan 15^o + \sqrt{3})^2 = 2^2 = 4 \]

Powered by Hirophysics.com