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. \(_\square\)
\[ \large\tan(x)+\sec(x)=2\cos(x)\]
Find the number of solutions of \(x\) in the interval \([0,2\pi] \) that satisfy the equation above.
\[ \large \color{purple}{\sin^{-1}} \left [ \color{blue} {\sin} (\color{green}{10}) \right ] = \, \color{brown}? \]
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. \(_\square \)
For \(x\in\mathbb{Z}\), find the probability that
\[2\sin x^{\circ}<1.\]
\[ A = \max_{x \in \mathbb{R}} \left( \log_2 3 \right)^{\sin x }, \qquad B = \max_{x \in \mathbb{R}} \left( \log_3 2 \right)^{\sin x }\]
Which is larger, \(A\) or \(B?\)
\[\large {f(x)= \displaystyle \lim_{n \to \infty} \dfrac{x}{(2\sin x)^{2n}+1}}\]
How many values of \(x\) are there from \(0\) to \(\frac{9\pi}{2}\) (both inclusive) such that \(f(x)\) is discontinuous at those values of \(x\).
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.
Find the total number of solutions of the equation
\[2x=3\pi(1-\cos x).\]
\[\large \prod_{r = 1}^{12} \sin (rx) = 0\]
What is the number of solutions of \(x\) satisfying the equation above in the interval \((0,\pi]?\)
How many real numbers \(x\) satisfy \(\sin x = \frac{x}{100}?\)
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.