Trigonometric Identities

Exercises

1. Determine the exact value, expressed in terms of radicals if necessary, for each of the following outputs of trigonometric functions.

   \(\displaystyle \cos(15°)\)

   \(\displaystyle \sin(15°)\)

   \(\displaystyle \tan(15°)\)

   \(\displaystyle \cos(7.5°)\)

   \(\displaystyle \sin(7.5°)\)

   \(\displaystyle \tan(7.5°)\)

   \(\displaystyle \cos(22.5°)\)

   \(\displaystyle \cos(11.25°)\)

   \(\displaystyle \cos(5.625°)\)

   \(\displaystyle \cos(2.8125°)\)

2. Write each of the following expression exclusively in terms of cosine.

   \(\displaystyle \sin(\theta) \)

   \(\displaystyle \tan(\theta) \)

   \(\displaystyle \sec(\theta)-\sin^2(\theta) \)

   \(\displaystyle \sin\Bigl(\frac{1}{2}\theta\Bigr) + \sin(2\theta) \)

3. Given that \(\cos(36°) = \frac{\sqrt{5}+1}{4},\) determine the exact value, expressed in terms of radicals if necessary, for each of the following outputs of trigonometric functions.

   \(\displaystyle \sin(36°) \)

   \(\displaystyle \cos(72°) \)

   \(\displaystyle \cos(18°) \)

   \(\displaystyle \cos(24°) \)

   \(\displaystyle \cos(9°) \)

   \(\displaystyle \cos(69°) \)

Problems & Challenges

1. Standing at a point on the earth, how far east of you is a point that is one mile northeast of you? How far east of you is a point that is one mile east-northeast of you? How far east of you is a point that is one mile east-by-north of you? How far east of you is a point that is one mile in the direction halfway between east-by-north and east of you?

   More abstractly, what is a formula, in terms of \(n,\) for \(\cos\bigl(\frac{\pi}{2^n}\bigr)?\)

2. Suppose \(\theta\) is some angle such that \(\sec(\theta) - \tan(\theta) = 2.\) What must the value of \(\sec(\theta) + \tan(\theta)\) be?
3. Suppose the cos and tan buttons on your calculator are broken — only sin works — but you need to know a decimal approximation for the value of \[\csc(42°)+\cos^2(42°)-\tan(42°).\] How do you calculate this number using only the sine function?
5. Andreescu Suppose a calculator is broken and the only keys that still work are the sin, cos, tan, sin^-1, cos^-1, and tan^-1 buttons. The calculator is powered on, and the display initially shows 0. Given any positive rational number \(q,\) show that we can get the decimal expansion of \(q\) to appear on the display panel of the calculator by pressing some finite sequence of those working buttons. Assume that the calculator does real-number calculations with infinite precision, and that all functions are defined in terms of radians.
6. What is the exact value, expressed in terms of radicals, of \(\cos\bigl(\frac{\pi}{5}\bigr)\) and \(\sin\bigl(\frac{\pi}{5}\bigr)?\)