三角比進階難解問題

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Question
Given that a sin x = b cos y = c tan z.
What are the possible values of a, b, c, x, y and z? Please verify your answer.

Solution
Consider
　a sin x = b cos y = c tan z = k
Then,
{ sin x = k/a
{ cos y = k/b
{ tan z = k/c

There are a lot of possible cases.

For example, if you force k = 1, you can consider
{ sin 30° = 1/2
{ cos 60° = 1/2
{ tan 45° = 1/1
One possible solution is
a = 2, b = 2, c = 1, x = 30°, y = 60°, z = 45°.

You can also consider
{ sin 30° = 1/2
{ cos 30° = √3/2 = 1/(2/√3)
{ tan 30° = 1/√3
Therefore, another possible solution is
a = 2, b = 2/√3, c = √3, x = 30°, y = 30°, z = 30°.

If you force k = √3, you can consider
{ sin 60° = √3/2
{ cos 30° = √3/2
{ tan 60° = √3/1
One possible solution is
a = 2, b = 2, c = 1, x = 60°, y = 30°, z = 60°.

You can also consider
{ sin 45° = √2/2 = 1/√2 = √3/√6
{ cos 60° = 1/2 = √3/(2√3)
{ tan 45° = 1 = √3/√3
Therefore, another possible solution is
a = √6, b = 2√3, c = √3, x = 45°, y = 60°, z = 45°.

There are many more possible solutions.

http://en.wikipedia.org/wiki/Trigonometric_functions#Special_values_in_trigonometric_functions


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