How to visualise positive and negative tangents

Question

A quick internet search of simple trigonometry methods returns a whole bunch of acronyms for remembering whether sin, cos and tan functions yield positive or negative results in the four quadrant of a circle.

I find this rather unsatisfactory, since it makes no attempt to explain what's going on.

To my mind, using a circle and lines (as below), it's fairly easy to see why, as the angle grows, the sine function returns positive results (red line) in the top half, and similarly the cosine (light blue line) goes negative in the left half of the circle and returns to positive as it comes back past 270°.

When I was taught trigonometry it was a complete mystery how the tan function related to the geometric tangent (my teacher told me "it was complicated"), and like everyone else simply memorised acronyms to get by. It now turns out its relationship is not complicated at all and a simple diagram like this goes a long way to explaining the mysteries of trig.

However, the part of the puzzle I'm missing regards the + and - of the tangent function. I understand how is the ratio of the geometric tangent (always returning to the x axis) to the 'radius'. (I now notice that the diagram is rather badly drawn in that respect.) What's not clear is why the tan of, say, 120° should be regarded as negative, but 200° a positive.

As with sine and cosine, is there a similarly simple way to visualise why it's negative in the top-left and bottom-right quadrants?

Answer 1

I think you have it backwards: $\tan \theta$ is negative in the top left and the bottom right quadrants.

As $\tan \theta$ is the ratio of $\sin \theta$ and $\cos \theta$, it depends on whether $\theta$ increases or decreases for their respective functions in that quadrant, and whether $\sin \theta$ and $\cos \theta$ are positive or negative (or both).

For instance, $\tan 120^\circ$ is negative because in the second quadrant, $\sin \theta$ is positive (but decreasing), while $\cos \theta$ is negative (but increasing). Thus $\tan \theta$ is negative in that quadrant.

In that same vein, $\tan 200^\circ$ is positive because in the third quadrant, $\sin \theta$ is negative (but decreasing), while $\cos \theta$ is negative (but increasing); thus $\tan \theta$ is positive in that quadrant.

Answer 2

$\sin \theta$ and $\cos \theta$ are the $y$- and $x$-coordinates of a point on the unit circle. So, sine is positive when $y$ is positive; for the top half of the unit circle. Likewise, cosine is positive when $x$ is positive; for the right half of the unit circle.

Since $\tan \theta = \frac{\sin \theta}{\cos \theta}$, note that the slope of the line connecting the origin to the point $(\cos \theta, \sin \theta)$ is exactly $\tan \theta$. So, tangent is positive when that line segment has positive slope (in the 1st and 3rd quadrants). I always think of tangent as a slope.

As an aside, a better visualization for $\tan \theta$ is to use the tangent to the circle at $(1,0)$. Extend the line connecting the origin to $(\cos \theta, \sin \theta)$ until it hits the line $x = 1$. This intersection occurs at the point $(1, \tan \theta)$. You can arrive at this yourself by scaling up the triangle with points $(0,0), (\cos \theta, \sin \theta)$, and $(\cos \theta, 0)$. The horizontal length is $\cos \theta$; to make it have length $1$, divide all the sides by $\cos \theta$

As yet another aside, you also see that this scaled-up triangle will have hypotenuse length $\sec \theta$; the word tangent comes from the Latin verb tangere meaning “to touch”, and gives the length of the line segment touching the circle at $(1,0)$. The word secant comes from the verb secare, meaning “to cut”. When you extend the line segment from the origin to $(\cos \theta, \sin \theta)$, it cuts through the circle and has length $\sec \theta$.

Answer 3

I think the $\tan$ function is typically defined this way for any $\theta$ such that $\cos\theta \neq 0.$ $$ \tan \theta = \frac{\sin \theta}{\cos \theta} $$ When $\cos\theta = 0,$ then $\tan\theta$ is undefined.

Defined this way, the "tangent" function is simply the slope of the radius from $(0,0)$ to $(\cos\theta, \sin\theta).$ For points in the first and third quadrants the slope is positive; in the other two quadrants it's negative.

I think this is very simple.

The definition above leaves open a big question, however: if $\tan\theta$ is the slope of a line, why do we call it a "tangent"? That question is what your figure answers.

Note that as long as we're only dealing with the interior angles of right triangles, we only need to be concerned with the first quadrant. The traditional names of trigonometric functions tend to be based on how we can interpret the functions with regard to those angles. Since your figure works perfectly well for all angles in the first quadrant, and since the tangent segment has the length $\tan\theta,$ "tangent" seems an appropriate name. (Actually, I think one usually sees a different figure in which the segment is tangent to the circle at $(1,0)$ and intersects the extended radius at $(1,\tan\theta),$ but the geometry works out the same.)