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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.

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  • $\begingroup$ Have you considered law of sines, law of cosines, etc.? $\endgroup$ Dec 2, 2023 at 2:18

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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.

Answer: 3960 miles

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.)

Answer: 92,800,000 miles

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.)

Answer: 433,000 miles

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.)

Answer: 4.63 light years

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  • $\begingroup$ indeed these sound quite interesting! $\endgroup$
    – athos
    Nov 25, 2020 at 8:22
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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.

enter image description here

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    $\begingroup$ indeed this is a smart construction, i'll show it to my kid, thank you! $\endgroup$
    – athos
    Nov 22, 2020 at 17:59
  • $\begingroup$ This is fantastic, it's interesting to know is there any way of doing such thing for other than acute angles of A and B. $\endgroup$ Dec 21, 2023 at 17:00
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I remember enjoying simplifying trig expressions. You can find exercises all over the web. One source: math-exercises.com. Here's a snippet:


Trig


Two more sources:
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  • $\begingroup$ For example, from the snippet: (a) $1+\cos x$, (b) $\cot^2 x$, (c) $-2 \cos x \sin x$. $\endgroup$ Nov 20, 2020 at 0:26
  • $\begingroup$ @joseph-orourke thank you very much, indeed these are quite interesting! $\endgroup$
    – athos
    Nov 20, 2020 at 10:29
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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!

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I would like to suggest following challenges,

  1. Geometric proof of sum and product of three tangents of a triangle are equal for obtuse-angled triangles.
    (https://www.janakasrodrigo.com/wp-content/uploads/2023/12/Proof-7-1.pdf)

  2. Trigonometric proof of Pythagoras theorem without applying properties of similar triangles and any form of Pythagorean Identity. (https://www.janakasrodrigo.com/wp-content/uploads/2023/12/Proof-4.pdf )
    Editor board of the journal American Mathematical Monthly acknowledged this as a new proof of Pythagoras theorem.

  3. Simultaneous single proof of the Sine Law and Cosine Law covering all three types of triangles.(https://www.janakasrodrigo.com/wp-content/uploads/2023/12/Proof-5.pdf)

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    $\begingroup$ pls how could you prove the first conclusion? $\endgroup$
    – athos
    Dec 22, 2023 at 2:18
  • $\begingroup$ @athos link of the requested proof added. $\endgroup$ Dec 23, 2023 at 16:05

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