Precalculus Flashcards: Extending Trigonometric Functions With Unit Circle

What this deck covers

This deck focuses on Extending Trigonometric Functions With Unit Circle, giving you a quick way to review the definitions, rules, and examples that matter most for Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: At which angles is $\tan\theta$ undefined on the unit circle?

Answer: $\theta=\frac{\pi}{2}+k\pi$, for integers $k$. Where $\cos\theta = 0$, making division undefined.

Flashcard 2: Identify the exact value of $\cos\left(-\frac{\pi}{3}\right)$ using unit-circle symmetry.

Answer: $\frac{1}{2}$. By even property: $\cos(-\frac{\pi}{3})=\cos(\frac{\pi}{3})=\frac{1}{2}$.

Flashcard 3: What is the reference angle for $\theta=\frac{7\pi}{6}$?

Answer: $\frac{\pi}{6}$. Reference angle is acute angle to nearest $x$-axis.

Flashcard 4: What is the symmetry identity for cosine that follows from the unit circle?

Answer: $\cos(-t)=\cos t$. Cosine is even: same $x$-coordinate for $\pm t$.

Flashcard 5: What is the relationship between arc length $s$ and angle $\theta$ on the unit circle?

Answer: $s=\theta$. On unit circle, arc length equals angle in radians.

Flashcard 6: Identify the period statement for sine and cosine using the unit circle.

Answer: $\sin(\theta+2\pi)=\sin\theta$ and $\cos(\theta+2\pi)=\cos\theta$. Functions repeat after one full rotation of $2\pi$.

Flashcard 7: What are the coordinates on the unit circle at $t=\frac{\pi}{2}$?

Answer: $(0,1)$. Quarter turn counterclockwise from $(1,0)$ reaches top of circle.

Flashcard 8: What point on the unit circle corresponds to angle $0$ radians?

Answer: $(1,0)$. Starting point where $\cos(0) = 1$ and $\sin(0) = 0$.

Flashcard 9: What is the period identity that extends sine to all real numbers $t$?

Answer: $\sin(t+2\pi)=\sin t$. Full rotation returns to same point, so sine repeats.

Flashcard 10: What is the unit circle in the coordinate plane?

Answer: The circle centered at $(0,0)$ with radius $1$. Defined by equation $x^2 + y^2 = 1$ in the coordinate plane.

Flashcard 11: Which direction corresponds to positive angles on the unit circle?

Answer: Counterclockwise. Standard convention: positive angles rotate counterclockwise.

Flashcard 12: What is the radian measure of a half revolution around the unit circle?

Answer: $\pi$. Half of the circumference $2\pi$ equals $\pi$.

Flashcard 13: What does it mean to interpret $t$ as a radian measure on the unit circle?

Answer: $t$ is the signed arc length traveled on the unit circle. Radians equal arc length on unit circle (radius = 1).

Flashcard 14: Identify the exact value of $\tan\left(\frac{\pi}{4}\right)$ using the unit circle.

Answer: $1$. At $\frac{\pi}{4}$, point is $(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$, so $\tan=\frac{y}{x}=1$.

Flashcard 15: Find the unit-circle point $(\cos\theta,\sin\theta)$ for $\theta=\pi$.

Answer: $(-1,0)$. Halfway around circle on negative $x$-axis.

Flashcard 16: What is the unit-circle definition of $\tan t$ when $\cos t \ne 0$?

Answer: $\tan t = \frac{\sin t}{\cos t}$. Ratio of $y$-coordinate to $x$-coordinate on the unit circle.

Flashcard 17: What is the direction of positive angle measure on the unit circle?

Answer: Counterclockwise from the positive $x$-axis. Standard convention for measuring positive angles.

Flashcard 18: What point on the unit circle corresponds to an angle (radian measure) of $t$?

Answer: $(\cos t, \sin t)$. Point reached by traveling $t$ radians counterclockwise from $(1,0)$.

Flashcard 19: What is the symmetry identity for sine that follows from the unit circle?

Answer: $\sin(-t)=-\sin t$. Sine is odd: opposite $y$-coordinates for $\pm t$.

Flashcard 20: Which trig functions are defined for every real number $\theta$ via the unit circle?

Answer: $\sin\theta$ and $\cos\theta$. Unit circle coordinates always exist for any angle.

Flashcard 21: Identify the exact value of $\sin\left(\frac{3\pi}{2}\right)$ using the unit circle.

Answer: $-1$. At $\frac{3\pi}{2}$, the point is $(0,-1)$, so $y=-1$.

Flashcard 22: What is the period identity that extends cosine to all real numbers $t$?

Answer: $\cos(t+2\pi)=\cos t$. Full rotation returns to same point, so cosine repeats.

Flashcard 23: Identify the exact value of $\cos(\pi)$ using the unit circle.

Answer: $-1$. At $\pi$ radians, the point is $(-1,0)$, so $x=-1$.

Flashcard 24: What is the definition of $\tan\theta$ using unit-circle coordinates?

Answer: $\tan\theta=\frac{\sin\theta}{\cos\theta}$ when $\cos\theta\ne 0$. Ratio of $y$-coordinate to $x$-coordinate on unit circle.

Flashcard 25: What ordered pair on the unit circle corresponds to angle $\theta$?

Answer: $(\cos\theta,\sin\theta)$. Point on unit circle at angle $\theta$ from positive $x$-axis.

Flashcard 26: Find the coterminal angle in $[0,2\pi)$ for $\theta=-\frac{\pi}{2}$.

Answer: $\frac{3\pi}{2}$. Add $2\pi$ to negative angle to find coterminal in $[0,2\pi)$.

Flashcard 27: Find $\sin\left(-\frac{\pi}{3}\right)$ using unit-circle symmetry.

Answer: $-\frac{\sqrt{3}}{2}$. Negative angle rotates clockwise; sine is odd function.

Flashcard 28: What is the definition of $\cos\theta$ using the unit circle?

Answer: $\cos\theta$ is the $x$-coordinate on the unit circle. Horizontal distance from origin to point on unit circle.

Flashcard 29: Which direction corresponds to negative angles on the unit circle?

Answer: Clockwise. Negative angles rotate in the opposite direction.

Flashcard 30: Find $\cos\left(\frac{5\pi}{3}\right)$ using the unit circle.

Answer: $\frac{1}{2}$. At $\frac{5\pi}{3}$, point is in fourth quadrant with $x = \frac{1}{2}$.

Flashcard 31: What is the definition of the unit circle in the coordinate plane?

Answer: Circle centered at $(0,0)$ with radius $1$. Standard form: center at origin with all points distance 1 away.

Flashcard 32: What is the radian measure of a quarter revolution around the unit circle?

Answer: $\frac{\pi}{2}$. One-fourth of the circumference $2\pi$ equals $\frac{\pi}{2}$.

Flashcard 33: What is the radian measure of one full revolution around the unit circle?

Answer: $2\pi$. Circumference of unit circle is $2\pi r = 2\pi(1) = 2\pi$.

Flashcard 34: What is the definition of $\sin\theta$ using the unit circle?

Answer: $\sin\theta$ is the $y$-coordinate on the unit circle. Vertical distance from origin to point on unit circle.