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April 13th, 2013, 03:01 PM   #1
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Which line segment equals tangent of angle

The answer is the tangent from the point of tangency to the x-axis equals the tangent of the angle (got that?)
Note: There is no question in here, no need to reply. I had this question on a test I just took and wanted to find out the answer. By the time I'd figured it out, I didn't want to lose all that I'd typed in here, so I leave it here for anyone's edification.

[attachment=0:2zpfaohr]Capture.PNG[/attachment:2zpfaohr]
The first quadrant of the unit circle is shown with a radius to point C $(a,b)$. The angle between the radius and the x-axis is $\theta$. The tangent to the circle at point C is drawn between the x-axis (point D) and y-axis (point E). The question asks which line segment is equal to the tangent of the labeled angle.

The answer is the segment from the point of tangency to the x-axis ($\overline{CD}$). Following is the proof

The, $\tan (\theta)= \frac ba$. So we need a segment which is $\frac ba$ units long. That is, I want to show that $\overline{CD}$ is $\frac ba$ units long.

The slope of the radius is also $\frac ba$, so the slope of the tangent (which is perpendicular) is $-\frac ab$. Since $(a,b)$ is on the line, we can find the equation of the line:
$y-b = -\frac ab \left(x-a\right) \\
y = -\frac ab \cdot x + \left(\frac {a^2 + b^2}{b} \right) \\
y = -\frac {a}{b} x + \frac {1}{b} \\$

The last step uses that $a^2 + b^2=1$ for any point $(a,b)$ on the unit circle. Then the x-intercept of the tangent is $\left( \frac 1a, 0 \right)$.

We need to show that the distance from $\left(a,b\right)$ to $\left(\frac 1a,0\right)$ equals $\frac ba$ (don't forget that $a^2 + b^2=1$):

$d= \sqrt{b^2 + \left( a - \frac 1a \right)^2} = \sqrt{b^2 + a^2 -2 + \frac 1{a^2}} = \sqrt{\frac 1{a^2} - 1} = \sqrt{\frac{1-a^2}{a^2}} = \sqrt{\frac{b^2}{a^2}} = \frac ba. \quad \qed$
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 April 13th, 2013, 07:40 PM #2 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 Re: Which line segment equals tangent of angle Another approach would be to use right triangle OCD and tanx = opposite/adjacent Since the adjacent side equals 1 then tanx equals the length of the opposite.
April 14th, 2013, 05:38 AM   #3
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Re: Which line segment equals tangent of angle

Nice! Thanks. (Hopefully, your reply appears before my post when people search for this.)
Quote:
 Originally Posted by agentredlum Another approach would be to use right triangle OCD and tanx = opposite/adjacent Since the adjacent side equals 1 then tanx equals the length of the opposite.

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