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The question is $x = \arctan\frac 23 + \arctan\frac 12$. What is $\tan(x)$?

I'm having trouble figuring out how to calculate the arctan values without a calculator, or do I not even need to find these values to calculate what $\tan(x)$ is? Any help is greatly appreciated (I would show some sort of work, but I am actually completely stuck).

I know that the range of arctan is restricted to $(–90^\circ, 90^\circ)$ or $(-\pi/2, \pi/2)$, but i'm not sure how this helps.

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3 Answers 3

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Hint

Use the formula $\tan(a+b)=\frac{\tan(a)+\tan(b)}{1-\tan(a)\tan(b)}$

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Let $A=\arctan\frac 23,B=\arctan\frac 12$.

Then, $$\tan x=\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}=\frac{\frac 23+\frac 12}{1-\frac 23\cdot \frac 12}.$$

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Below is a solution based on right triangles.

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Note that $y=\arctan\frac{2}{3}$, and $z=\arctan\frac{1}{2}$. We note that $x = y + z$.

This is because the sum of the two angles next to $x$ along with $x$ itself is 180 degrees. But the angle to the left side of $x$ summed with $y$ is 90 degrees. And the angle to the right side of $x$ summed with $z$ is also 90 degress.

The question is then: Calculate $\frac{b}{a}$.

Since $a^2 + c^2 = 4$ and $\frac{a}{c} = \frac{2}{3}$, it follows that $a^{2} = \frac{16}{13}$.

Since $a^2 + b^2 = 5$, it follows that $b^2 = \frac{49}{13}$.

Thus, $\frac{b}{a} = \frac{7}{4}$.


The take-away should be that sometimes going to a more basic line of thought can get you unstuck on a problem. Also, this gives some insight on how you can derive the tangent of sums formula.

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