Sine and Cosine Function Values - AP Precalculus

1. Which is greater: $\text{sin}(\frac{\text{π}}{4})$ or $\text{cos}(\frac{\text{π}}{4})$?

   Equal. Both equal $\frac{\sqrt{2}}{2}$ at the 45° angle.

2. What is the value of $\text{cos}(\text{π})$?

   -1. At $\pi$, we're at the leftmost point where x-coordinate is -1.

3. What is the value of $\text{sin}(\frac{3\text{π}}{2})$?

   -1. At $\frac{3\pi}{2}$, we're at the bottom of the unit circle.

4. Evaluate $\text{cos}(270^\text{o})$.

   0. 270° is equivalent to $\frac{3\pi}{2}$ radians.

5. What is the phase shift of $y = \text{cos}(x + \frac{\text{π}}{3})$?

   $\frac{\text{π}}{3}$ to the left. Horizontal shift is $-\frac{C}{B}$ where $C=-\frac{\pi}{3}$ and $B=1$.

6. Find $\sin(-\pi)$ using symmetry.

   0. Sine is an odd function: $\sin(-x) = -\sin(x)$.

7. What is the value of $\text{sin}(\text{π})$?

   0. At $\pi$, we're at the leftmost point where y-coordinate is zero.

8. Evaluate $\text{cos}(90^\text{o})$.

   0. 90° is equivalent to $\frac{\pi}{2}$ radians.

9. What is the cosine of a 45-degree angle?

   $\frac{\sqrt{2}}{2}$. 45° corresponds to $\frac{\pi}{4}$ radians on the unit circle.

10. Identify the range of $y = \text{cos}(x)$.

    [-1, 1]. Cosine oscillates between its maximum and minimum values.

11. Identify the range of $y = \text{sin}(x)$.

    [-1, 1]. Sine oscillates between its maximum and minimum values.

12. Find $\text{sin}(-\text{π})$ using symmetry.

    0. Sine is an odd function: $\sin(-x) = -\sin(x)$.

13. Evaluate $\sin\left(\frac{\pi}{3}\right)$.

    $\frac{\sqrt{3}}{2}$. 60° angle corresponds to the longer leg in a 30-60-90 triangle.

14. What is the reciprocal of $\sin(x)$?

    $\csc(x)$. Cosecant is defined as $\frac{1}{\sin(x)}$.

15. What is the amplitude of $y = -2\text{cos}(x)$?

    2. Amplitude is the absolute value of the coefficient of cosine.

16. What is the value of $\text{cos}(\frac{3\text{π}}{2})$?

    0. At $\frac{3\pi}{2}$, the x-coordinate is zero.

17. What is the sine of a 45-degree angle?

    $\frac{\sqrt{2}}{2}$. 45° corresponds to $\frac{\pi}{4}$ radians on the unit circle.

18. Evaluate $\sin(\frac{\pi}{3})$.

    $\frac{\sqrt{3}}{2}$. 60° angle corresponds to the longer leg in a 30-60-90 triangle.

19. What is the period of $y = \cos(\frac{x}{2})$?

    $4\pi$. Period equals $\frac{2\pi}{|B|}$ where $B=\frac{1}{2}$.

20. What is the phase shift of $y = \text{sin}(x - \frac{\text{π}}{4})$?

    $\frac{\text{π}}{4}$ to the right. Horizontal shift is $\frac{C}{B}$ where $C=\frac{\pi}{4}$ and $B=1$.

21. What is the period of $y = \text{cos}(\frac{x}{2})$?

    $4\text{π}$. Period equals $\frac{2\pi}{|B|}$ where $B=\frac{1}{2}$.

22. Evaluate $\text{cos}(360^\text{o})$.

    1. 360° is equivalent to $2\pi$ radians.

23. Evaluate $\text{sin}(270^\text{o})$.

    -1. 270° is equivalent to $\frac{3\pi}{2}$ radians.

24. What is the reciprocal of $\text{cos}(x)$?

    $\text{sec}(x)$. Secant is defined as $\frac{1}{\cos(x)}$.

25. Evaluate $\text{cos}(180^\text{o})$.

    -1. 180° is equivalent to $\pi$ radians.

26. Evaluate $\text{sin}(\frac{\text{π}}{6})$.

    $\frac{1}{2}$. 30° angle gives the smallest positive sine value in special triangles.

27. What is the value of $\text{sin}(2\text{π})$?

    0. After one complete revolution, we return to the starting y-coordinate.

28. Evaluate $\text{sin}(360^\text{o})$.

    0. 360° is equivalent to $2\pi$ radians.

29. What is the value of $\text{sin}(0)$?

    0. At the origin, sine is the y-coordinate on the unit circle.

30. What is the frequency of $y = \cos(5x)$?

    $\frac{5}{2\pi}$. Frequency is $\frac{|B|}{2\pi}$ where $B=5$.

31. What is the period of $y = \text{sin}(2x)$?

    $\text{π}$. Period equals $\frac{2\pi}{|B|}$ where $B=2$.

32. What is the value of $\text{cos}(0)$?

    1. At the origin, cosine is the x-coordinate on the unit circle.

33. Evaluate $\text{cos}(\frac{\text{π}}{3})$.

    $\frac{1}{2}$. 60° angle gives the smallest positive cosine value in special triangles.

34. What is the value of $\text{cos}(\frac{\text{π}}{2})$?

    0. At $\frac{\pi}{2}$, the x-coordinate is zero.

35. Evaluate $\cos\left(\frac{\pi}{6}\right)$.

    $\frac{\sqrt{3}}{2}$. 30° angle corresponds to the longer leg in a 30-60-90 triangle.

36. What is the value of $\text{cos}(2\text{π})$?

    1. After one complete revolution, we return to the starting x-coordinate.

37. What is the amplitude of $y = 3\text{sin}(x)$?

    3. Amplitude is the absolute value of the coefficient of sine.

38. Find $\text{cos}(-\text{π})$ using symmetry.

    -1. Cosine is an even function: $\cos(-x) = \cos(x)$.

39. What is the period of $y = \sin(2x)$?

    $\pi$. Period equals $\frac{2\pi}{|B|}$ where $B=2$.

40. Evaluate $\text{sin}(90^\text{o})$.

    1. 90° is equivalent to $\frac{\pi}{2}$ radians.

41. What is the value of $\text{cos}(\text{π} - x)$?

    $-\text{cos}(x)$. Uses the cosine supplementary angle identity.

42. What is the value of $\text{sin}(\frac{\text{π}}{2})$?

    1. At $\frac{\pi}{2}$, we reach the top of the unit circle.

43. Evaluate $\text{sin}(180^\text{o})$.

    0. 180° is equivalent to $\pi$ radians.

Study these flashcards