All trig functions in one figure


Here’s a pretty thing that got completely missed in my grammar school mathematics class. We build a geometric diagram in which each trigonometric function corresponds to the length of a line.

Remember the trigonometric functions? You’ll need them to appreciate what I’m going to show you. If you’ve forgotten, here is a review. If you remember trig functions (sine, cosine, tangent, secant, cosecant and cotangent) then skip this section.

Review of trig functions

Any angle θ (Greek letter ‘theta’) between two straight lines generates six useful numbers: the sine ( abbreviated sin(θ) ), the cosine ( cos (θ) ), the tangent ( tan (θ) ), the secant ( sec (θ) ), the cosecant ( csc (θ) ) and the cotangent ( cot (θ) ).

We can calculate these six numbers by making θ the base angle of a right-angled triangle.

In this diagram, θ is the size of angle BAC. Angle BCA is the right angle.

The three most used trig functions are the sine, cosine and tangent.

\sin(\theta)=\frac {a} {c}

\huge{\cos(\theta)=\frac {b} {c}}

\tan(\theta)=\frac {a} {b}

The other three functions are the reciprocals of the sine, cosine and tangent.

csc(\theta) = \frac{1}{sin(\theta)} = \frac {c} {a}

sec(\theta) = \frac{1}{cos(\theta)} = \frac {c} {b}

cot(\theta) = \frac{1}{tan(\theta)} = \frac {b} {a}

End of review; on with the good stuff

These ratios are the same no matter how big or small the triangle is. Increase each of the sides by a factor of 10, for example, and

sin(\theta) = \frac {10 a} {10c} = \frac {a} {c} ,

same as before.

That last observation means that we can insist that c = 1 and so long as θ remains the same, the trigonometric functions have the same values.

In fact, if c = 1, then

\huge \sin(\theta)=\frac{a} {1} = a

and

cos(\theta) = \frac{b} {1} = b

So the two sides BC and AC have lengths sin(θ) and cos(θ) respectively.

Now take the triangle with c = 1 and draw a circle around it, a circle with centre at A and radius 1. A circle with radius equal to 1 is called a unit circle.

Now draw the tangent to the circle at B and extend AC to meet the circle at D and the tangent at E.

Since angle ABE is a right angle (a tangent always makes a right angle with the radius), triangle AEB is similar to triangle ABC. Specifically

angle EAB = angle BAC  (= θ)

angle ABE = angle ACB (both right angles)

so angle ABE = angle ABC

So, from triangle AEB, we get the following.

tan (\theta) = \frac{BE} {BA} = \frac{BE} {1} = BE

sec (\theta) = \frac{AE} {BA} = AE

Finally, draw a line segment from A parallel to CB that meets the tangent at F.

This creates a triangle AFB with a right angle at B, hypotenuse AF and angle AFB equal to θ. It takes one line of proof to establish that last fact. In this triangle

csc(\theta)= \frac {AF} {AB}=AF

and

cot(\theta) = \frac {BF} {AB} = BF

In our complete diagram, therefore, the value of each trigonometric function is represented by a line of that length. Very Greek.

This was completely new to me when I discovered it recently. Do most people with a high school trig course behind them know this?

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About aharmlessdrudge

Way back during the late Bronze age -- actually it was the 1950s -- all of us in high school had to take a vocational test to determine our interests and, supposedly, our future careers. I cannot remember the outcome, but I do recall one question that gave me pause. "If you were to win a Nobel prize, would it be in literature or in physics?" I hesitated over the question: although I enjoyed mathematics and science more than English class, I did have a couple of unfinished (and very bad) novels hidden away at home. I cannot remember what I chose back then, but the dilemma followed me to university, where I switched from mathematics to English and -- after a five-year stint in journalism -- back to mathematics. I recently retired as a professor of statistics. Retirement. What a good chance to revive my literary ambitions. I have finished a novel -- more about that in good time -- and a rubble of drafts of articles about mathematics and statistics is taking up space on my hard disk.
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