Graphs of Trigonometric Functions - Problem Solving

To solve the problems on this page, you should be familiar with the following:

解决问题——基本

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. $_\square$

3\pi-10

10-3\pi

10

3\pi+10

$\large \color{#69047E}{\sin^{-1}} \left [ \color{#3D99F6} {\sin} (\color{#20A900}{10}) \right ] = \, \color{#624F41}?$

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. $_\square$

A B Both are equal

$A = \max_{x \in \mathbb{R}} \left( \log_2 3 \right)^{\sin x }, \qquad B = \max_{x \in \mathbb{R}} \left( \log_3 2 \right)^{\sin x }$

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 $\frac{9\pi}{2}$ (包容) $f(x)$ is discontinuous at those values of $x$ .

Problem Solving - Advanced

B < C< A < D

D < A < B < C

A < D < C < B

C < B < D < A

$A = \sin\left[\sin(1)\right] \\ B = \sin\left[\cos(1)\right] \\ C = \cos\left[\sin(1)\right] \\ D =\cos\left[\cos(1)\right]$

The above are the values of $A,B,C,$ and $D.$ Which of the answer choices is true?

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 $f_k(\alpha )\neq \alpha?$

Details and Assumptions:The function is evaluated in radians. There is no degree symbol in the problem.

Quiz

Relevant For...

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