Extending Trigonometric Functions with Unit Circle - Precalculus

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

10. What is the unit circle in the coordinate plane?

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

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

    Counterclockwise. Standard convention: positive angles rotate counterclockwise.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    Clockwise. Negative angles rotate in the opposite direction.

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

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

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

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

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

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

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

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

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

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

Study these flashcards