Sine, Cosine, and Tangent - AP Precalculus

1. Find $\text{tan}(\frac{\pi}{4})$ using the unit circle.

   1. This is a standard angle value on the unit circle.

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

   -1. At $\pi$, the point is $(-1, 0)$ on the unit circle.

3. What is the cosine of $60^\text{o}$?

   $\frac{1}{2}$. From the $30°-60°-90°$ triangle or unit circle.

4. Convert $\frac{\pi}{3}$ radians to degrees.

   $60^\circ$. Use the formula $\text{degrees} = \text{radians} \times \frac{180^\circ}{\pi}$

5. What is the period of the tangent function?

   $\pi$. Tangent repeats its values every $\pi$ radians.

6. What is the tangent of $45^\text{o}$?

   1. In a $45°-45°-90°$ triangle, opposite equals adjacent.

7. What is the general solution for $\text{sin}(x) = 0$?

   $x = n\text{\textpi}$, where $n \text{ is an integer}$. Sine equals zero at integer multiples of $\pi$.

8. Find the cosine of angle $B$ if $\text{sin}(B) = \frac{5}{13}$ and $B$ is in Quadrant I.

   $\frac{12}{13}$. Use $\sin^2 B + \cos^2 B = 1$ to find $\cos B = \sqrt{1 - (\frac{5}{13})^2}$.

9. Convert $45^\text{o}$ to radians.

   $\frac{\text{\textpi}}{4}$. Use the formula $\text{radians} = \text{degrees} \times \frac{\pi}{180°}$.

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

    $\frac{\pi}{4}$ to the right. A negative value inside parentheses shifts the graph right.

11. Which quadrant has positive sine and cosine values?

    Quadrant I. All trigonometric functions are positive in Quadrant I.

12. What is the value of $\tan(\pi)$?

    0. Since $\tan(\pi) = \frac{\sin(\pi)}{\cos(\pi)} = \frac{0}{-1}$.

13. What is the phase shift of $y = \cos(x + \frac{\pi}{6})$?

    $\frac{\pi}{6}$ to the left. A positive value inside parentheses shifts the graph left.

14. What is the reciprocal of tangent?

    Cotangent. Cotangent is $\frac{1}{\tan(x)}$.

15. What is the sine of $30^\text{o}$?

    $\frac{1}{2}$. From the $30°-60°-90°$ triangle or unit circle.

16. Find $\sin\left(\frac{\pi}{6}\right)$ using the unit circle.

    $\frac{1}{2}$. This is a standard angle value on the unit circle.

17. State the definition of tangent in a right triangle.

    Tangent is the ratio of the opposite side to the adjacent side. This is the basic TOA definition from SOH-CAH-TOA.

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

    $\frac{1}{2}$. This is a standard angle value on the unit circle.

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

    3. The coefficient of the sine function is the amplitude.

20. What is the general solution for $\sin(x) = 0$?

    $x = n\pi$, where $n \text{ is an integer}$. Sine equals zero at integer multiples of $\pi$.

21. What is the tangent of $0^\text{o}$?

    0. Since $\tan(0^\circ) = \frac{\sin(0^\circ)}{\cos(0^\circ)} = \frac{0}{1}$.

22. What is the value of $\cos(\pi)$?

    -1. At $\pi$, the point is $(-1, 0)$ on the unit circle.

23. State the definition of cosine in a right triangle.

    Cosine is the ratio of the adjacent side to the hypotenuse. This is the basic CAH definition from SOH-CAH-TOA.

24. State the Pythagorean identity involving sine and cosine.

    $\text{sin}^2 \theta + \text{cos}^2 \theta = 1$. This comes from the Pythagorean theorem applied to the unit circle.

25. What is the reciprocal of cosine?

    Secant. Secant is $\frac{1}{\cos(x)}$.

26. Convert $\frac{\text{\textpi}}{3}$ radians to degrees.

    $60^\text{o}$. Use the formula $\text{degrees} = \text{radians} \times \frac{180°}{\pi}$.

27. What is the tangent of an angle in Quadrant IV?

    Negative. Tangent is negative in Quadrants II and IV.

28. What is the value of $\text{sin}(\text{\textpi})$?

    0. At $\pi$, the point is $(-1, 0)$ on the unit circle.

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

    2. The absolute value of the coefficient is the amplitude.

30. Find $\tan(\frac{\pi}{4})$ using the unit circle.

    1. This is a standard angle value on the unit circle.

31. Find the tangent of angle $C$ if $\text{sin}(C) = \frac{3}{5}$ and $C$ is in Quadrant I.

    $\frac{3}{4}$. First find $\cos C = \frac{4}{5}$, then $\tan C = \frac{\sin C}{\cos C}$.

32. What is the period of the cosine function?

    $2\pi$. Cosine repeats its values every $2\pi$ radians.

33. What is the vertical shift of $y = \text{tan}(x) + 3$?

    3 units up. Adding a constant outside the function shifts vertically.

34. What is the period of the sine function?

    $2\pi$. Sine repeats its values every $2\pi$ radians.

35. What is the value of $\text{tan}(\text{\textpi})$?

    0. Since $\tan(\pi) = \frac{\sin(\pi)}{\cos(\pi)} = \frac{0}{-1}$.

36. What is the reciprocal of sine?

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

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

    $\frac{\pi}{6}$. Reference angle is $\frac{7\pi}{6} - \pi = \frac{\pi}{6}$.

38. What is the cosine of an angle in Quadrant III?

    Negative. Cosine is negative in Quadrants II and III.

39. Find the sine of angle $A$ if $\text{cos}(A) = \frac{3}{5}$ and $A$ is in Quadrant I.

    $\frac{4}{5}$. Use $\sin^2 A + \cos^2 A = 1$ to find $\sin A = \sqrt{1 - (\frac{3}{5})^2}$.

40. What is the cosine of $0^\text{o}$?

    1. At $0°$, the point is $(1, 0)$ on the unit circle.

41. What is the reference angle for $210^\text{o}$?

    $30^\text{o}$. Reference angle is $210° - 180° = 30°$.

42. What is the sine of $90^\text{o}$?

    1. At $90°$, the point is $(0, 1)$ on the unit circle.

43. State the definition of sine in a right triangle.

    Sine is the ratio of the opposite side to the hypotenuse. This is the basic SOH definition from SOH-CAH-TOA.

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