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Precalculus Trigonometry Flashcards
50 question-and-answer cards covering Trigonometry as it is examined in Precalculus. 24 of them are printed below, taken from across the deck — no signup, no paywall on the preview.
24 sample cards from the Trigonometry deck
Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.
What is the sign of $\tan\theta$ in each of the four quadrants?
Positive in Quadrants I and III (where $x$ and $y$ have the same sign); negative in Quadrants II and IV (where $x$ and $y$ have opposite signs).
Define a reference angle.
A reference angle is the acute angle (between $0$ and $\frac{\pi}{2}$) formed between the terminal side of $\theta$ and the horizontal $x$-axis.
How do you find the reference angle for an angle $\theta$ in each quadrant (for $0 \le \theta < 2\pi$)?
QI: $\theta$; QII: $\pi - \theta$; QIII: $\theta - \pi$; QIV: $2\pi - \theta$.
How are reference angles used to find exact trig values of non-acute angles?
Evaluate the trig function of the reference angle, then attach the correct sign based on the quadrant of $\theta$ (using the ASTC rule).
What are the amplitude, period, and range of $y = \sin x$ and $y = \cos x$?
Amplitude $= 1$, period $= 2\pi$, and range $= [-1, 1]$ for both.
For $y = a\sin(bx)$ or $y = a\cos(bx)$, what are the amplitude and period?
Amplitude $= |a|$ and period $= \frac{2\pi}{|b|}$.
Compare the graphs of $y = \sin x$ and $y = \cos x$ in terms of a horizontal shift.
The cosine graph is the sine graph shifted left by $\frac{\pi}{2}$: $\cos x = \sin\left(x + \frac{\pi}{2}\right)$. Sine passes through the origin (odd); cosine has a maximum at $x=0$ (even).
What is the period of $y = \tan x$ and where are its vertical asymptotes?
Period $= \pi$; vertical asymptotes occur where $\cos x = 0$, i.e., at $x = \frac{\pi}{2} + \pi k$ for integer $k$.
What is the period of $y = \cot x$ and where are its vertical asymptotes?
Period $= \pi$; vertical asymptotes occur where $\sin x = 0$, i.e., at $x = \pi k$ for integer $k$.
Compare the behavior of $y = \tan x$ and $y = \cot x$ between consecutive asymptotes.
$\tan x$ is increasing on each interval between its asymptotes; $\cot x$ is decreasing on each interval between its asymptotes.
Where are the vertical asymptotes of $y = \sec x$ and what is its range?
Asymptotes where $\cos x = 0$, i.e., $x = \frac{\pi}{2} + \pi k$; range is $(-\infty, -1] \cup [1, \infty)$.
Where are the vertical asymptotes of $y = \csc x$ and what is its range?
Asymptotes where $\sin x = 0$, i.e., $x = \pi k$; range is $(-\infty, -1] \cup [1, \infty)$.
What are the period of $y = \sec x$ and $y = \csc x$?
Both have period $2\pi$, matching their reciprocal functions cosine and sine.
What is the domain and range of $y = \sin^{-1} x$ (arcsine)?
Domain $[-1, 1]$; range $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$.
What is the domain and range of $y = \cos^{-1} x$ (arccosine)?
Domain $[-1, 1]$; range $[0, \pi]$.
What is the domain and range of $y = \tan^{-1} x$ (arctangent)?
Domain $(-\infty, \infty)$; range $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.
Why must the domains of sine, cosine, and tangent be restricted to define their inverses?
The trig functions are periodic and therefore not one-to-one; restricting the domain to an interval where each is one-to-one allows a well-defined inverse that passes the horizontal line test.
On what restricted domain is $\sin x$ made one-to-one to define arcsine?
$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, where sine increases from $-1$ to $1$.
On what restricted domain is $\cos x$ made one-to-one to define arccosine?
$[0, \pi]$, where cosine decreases from $1$ to $-1$.
Evaluate $\sin^{-1}\left(\frac{1}{2}\right)$, $\cos^{-1}(0)$, and $\tan^{-1}(1)$.
$\sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6}$, $\cos^{-1}(0) = \frac{\pi}{2}$, $\tan^{-1}(1) = \frac{\pi}{4}$.
Simplify $\sin\left(\sin^{-1} x\right)$ and state its valid domain.
$\sin\left(\sin^{-1} x\right) = x$ for all $x \in [-1, 1]$.
Does $\sin^{-1}(\sin x) = x$ for all $x$? Explain.
No. $\sin^{-1}(\sin x) = x$ only when $x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. Outside that interval it returns the angle in the range of arcsine that has the same sine value.
Simplify $\cos\left(\sin^{-1} x\right)$ to an algebraic expression.
$\cos\left(\sin^{-1} x\right) = \sqrt{1 - x^{2}}$, using a right triangle with opposite $x$ and hypotenuse $1$.
Simplify $\tan\left(\cos^{-1} x\right)$ to an algebraic expression.
$\tan\left(\cos^{-1} x\right) = \frac{\sqrt{1 - x^{2}}}{x}$, from a right triangle with adjacent $x$ and hypotenuse $1$ (so opposite $= \sqrt{1-x^{2}}$).
What this deck covers
The Trigonometry deck follows the Precalculus Trigonometry syllabus — 5 chapters and 15 topics — so questions land on material that is genuinely examinable rather than trivia around it. That works out to roughly 10.0 cards per chapter.
Answers are written to be recallable, not just readable — averaging about 107 characters, which is long enough to carry the reasoning and short enough to say out loud.
A deck like this earns its keep on the second and third pass. Read the syllabus first so you know the shape of the subject, then use the cards to find the specific facts that have not stuck.
Trigonometry flashcards FAQ
How many Trigonometry flashcards are in this Precalculus deck?
50 cards. This page previews 24 of them, sampled evenly across the deck so you can judge the difficulty before installing anything.
Are these Precalculus flashcards free?
Yes. The preview here is free to read with no signup, and the full 50-card deck is free inside the Examius app.
What do the Trigonometry cards cover?
They follow the Precalculus Trigonometry syllabus — 5 chapters and 15 topics — so the questions track what is actually examinable.
How should I use these flashcards?
Read the syllabus first so you know the shape of the subject, then drill the deck. Examius schedules each card with spaced repetition, so cards you keep missing come back sooner and ones you know drift further apart.