Graphs of Trigonometric Functions - Problem Solving
To solve the problems on this page, you should be familiar with the following:
Problem Solving - Basic
What is the fundamental period of the function \(f(x) = \cos^2\big(\sin(x)\big)? \)
Note that by graphing, we can see that \( \cos^2\big(\sin(x)\big) = \cos^2\big(\sin(x+ k\pi)\big) \) holds true for all integer \(k\). Thus the fundamental period of \(f(x) \) is \(\pi\). \(_\square\)
P.S. We can also solve this via compound angle formula.
Evaluate \(\sin \big(10^{10}\big)^\circ \div \sin 80^\circ.\)
Because \(y = \sin x \) has a fundamental period of 360 degrees,
\[\sin \big(10^{10}\big)^\circ = \sin \big(10^{10} \bmod{360}\big)^\circ = \sin 280^\circ = -\sin 80^\circ. \]
Thus taking the quotient yields \(-1\) as the answer. \(_\square\)
Problem Solving - Intermediate
Given that \(\tan 1^\circ > \frac1{90},\) which of these numbers is larger, \(\tan(\tan 1^\circ)\) or \(\tan(\cot 1^\circ)?\)
Because \(\cot 1^\circ \) is the reciprocal of \(\tan 1^\circ, \) which is rather small, \( \tan 1^\circ < \cot 1^\circ.\) Also, because both of them are positive and less than 90, they are in the first quadrant. With \(y = \tan x \) as an increasing function, we have \( \tan( \tan 1^\circ) < \tan(\cot 1^\circ). \) So the latter number is larger. \(_\square \)
Problem Solving - Advanced
\[ A = \sin\left[\sin(1)\right] \\ B = \sin\left[\cos(1)\right] \\ C = \cos\left[\sin(1)\right] \\ D =\cos\left[\cos(1)\right] \]
The above are the values of \(A,B,C,\) and \(D.\) Which of the answer choices is true?
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Clarification: All angles are measured in radians.
Inspiration.
For every integer \(k,\) we define a function \(f_k \) by the formula
\[f_k(x)=100x-k\sin x.\]
What is the smallest positive integer value of \(k\) such that, for some real \(\alpha\), we have \(f_k\big(f_k(\alpha)\big)=\alpha,\) but \(f_k(\alpha )\neq \alpha?\)
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Details and Assumptions: The function is evaluated in radians. There is no degree symbol in the problem.