The man on the right is Bill Cosby, an actor and comedian who became a staple of Thursday night television for NBC from the mid-1980’s to the early 1990’s. (More recently, he has become embroiled in controversy because of allegations regarding extremely unsavory actions in his past. That’s not relevant to this joke, but it probably limits the joke’s “re-tell” value.) His last name is pronounced “KAHZ-bee”, which is important to know if you want to get the joke.
The rest of this image is all about trigonometry. You remember trigonometry, right? Maybe? No? Okay, let’s have a refresher.
Imagine a right triangle. A right triangle has one ninety-degree angle, and two angles that are both less than 90º. Let’s call these angles a, b, and c, with c being the right angle.
Now for the sake of convenience, I’m going to label the sides A, B, and C. Side A will be directly opposite angle a, and so on.
Side C, which is opposite angle c, is the hypotenuse. The hypotenuse of a right triangle is always opposite the 90º angle, and it is always the longest of the three sides.
Sides A and B are known as legs. There is a web of relationships between the measures of angles a, b, and c, and the lengths of sides A, B, and C.
Consider angle b, in the lower left corner of the triangle.
From the perspective of angle b, side B is opposite and side A is adjacent, meaning that side A is one of the legs that forms angle b. Side C is the hypotenuse, as always, and also encloses angle b. Clear as mud? Good.
To fully understand this joke, you need to know about three basic trigonometric functions: sine, cosine, and tangent. These functions are often abbreviated sin, cos, and tan. The sine of an angle is found by dividing the length of the opposite side by the length of the hypotenuse. For angle b, that’s side B divided by side C.
The cosine of angle b is found by dividing the length of the adjacent side (side A) by the length of the hypotenuse (side C).
The tangent of angle b is found by dividing the length of the opposite side (side B) by the length of the adjacent side (side A). It does not involve the hypotenuse at all.
There’s another interesting relationship between sine, cosine, and tangent. The tangent of angle b is equal to sin b divided by cos b. I’ll leave it as an exercise for the reader to prove that.
Now we’re in the home stretch. Multiplying both sides by cos b gives:
And dividing both sides by tan b gives: