# Interesting Trigonometry problems

After explaining some basic trigonometry to my kid, such as $$\sin (\alpha+\beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$$, Law of sines, Law of cosines, I wonder if there are some interesting problems for him to work on? Even better if it's a book.

Example for "interesting", $$\sin 0 = \frac{\sqrt{0}}{2}$$, $$\sin \frac\pi 6 = \frac{\sqrt{1}}{2}$$, $$\sin \frac\pi 4 = \frac{\sqrt{2}}{2}$$, $$\sin \frac \pi 3 = \frac{\sqrt{3}}{2}$$, while $$\sin \frac\pi 2 = \frac{\sqrt{4}}{2}$$. Or more details in wiki page Trigonometric constants expressed in real radicals.

Explanation in searching of interesting problems -- I believe exercises are necessary for one to really get the ideas and tricks of trigonometry, but they might also be boring, so if there are a bunch of interesting problems, that'll be perfect.

There are many interesting trigonometry problems in "Trigonometry for the Practical Man" by James Edgar Thompson. Here are some from Chapter 5:

Problem 1

From the top of a mountain three miles above sea level, the angle of depression of the ocean horizon is found to be 2° 13' 50". Find the radius of the earth.

Problem 2

An observer sights a telescope on the sun, then turns the telescope straight up to the zenith, measuring an angle of 30° 0' 13.2". At this same moment (by previous agreement), another observer 8280 miles away reports the sun is just setting. Find the distance from the center of the earth to the center of the sun. (You will need the earth’s radius found in problem 1.)

Problem 3

From a point on the surface of the earth, the sun is sighted with a transit, first on one edge and then on the other, and the angle between the lines of sight is found to be 32' 4". Find the radius of the sun. (You will need the answers to problems 1 and 2.)

Problem 4

On January 1, an astronomer measures the sun-earth-star angle of the star Proxima Centauri, and finds it to be 95° 27' 45", and six months later measures the angle and finds it has changed to 84° 32' 13.6". Find the distance from the earth to the star on January 1. (You will need to use the earth-sun distance found in problem 2.)

• indeed these sound quite interesting! Nov 25, 2020 at 8:22

When introducing the sum of angle equations, I practiced presenting this. Starting with introducing a right triangle inside a rectangle. (I misplaced my notes, in which I had every step clearly laid out.) In the end, it was a great proof of both equations. • indeed this is a smart construction, i'll show it to my kid, thank you! Nov 22, 2020 at 17:59

I remember enjoying simplifying trig expressions. You can find exercises all over the web. One source: math-exercises.com. Here's a snippet: Two more sources:
• For example, from the snippet: (a) $1+\cos x$, (b) $\cot^2 x$, (c) $-2 \cos x \sin x$. Nov 20, 2020 at 0:26
• @joseph-orourke thank you very much, indeed these are quite interesting! Nov 20, 2020 at 10:29

You can use the double angle formulas (together with the small angle approximations $$\sin(\theta) \approx \theta$$ and $$\cos(\theta) \approx 1$$) to approximate sine and cosine of any number $$x$$ by using the small angle approximations for $$\frac{x}{2^n}$$ and doubling until you get $$x$$. This is a pretty cool "proto-calculus" way to compute sine and cosine using only hand calculations. Makes a nice coding project too!