Trigonometry - SAT Math

1. Find $\text{cos}(60^\text{o})$ using cofunction identity.

   $\text{sin}(30^\text{o}) = \frac{1}{2}$. Using $\cos(90° - 30°) = \sin 30°$.

2. Identify the reciprocal of tangent.

   Cotangent (cot). Cotangent is defined as $\frac{1}{\tan}$.

3. Identify the cofunction identity for sine.

   $\text{sin}(90^\text{o} - \theta) = \text{cos}(\theta)$. Complementary angle relationship for sine.

4. What is the sine of $60^{\text{o}}$?

   $\frac{\sqrt{3}}{2}$. Standard value for 60° in the unit circle.

5. What is the value of $\text{cos} 0^\text{°}$?

   1. At 0°, the point on the unit circle is (1,0), so cosine equals 1.

6. State the angle addition formula for tangent.

   $\text{tan}(A + B) = \frac{\text{tan}A + \text{tan}B}{1 - \text{tan}A \text{tan}B}$. Formula for tangent of sum of two angles.

7. Identify the reciprocal of sine.

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

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

   $\frac{1}{2}$. Standard value for 30° in the unit circle.

9. What is the range of the sine function?

   [-1, 1]. Sine values are bounded between -1 and 1.

10. What is the value of $\text{cos}^2 45^\text{o}$?

    $\frac{1}{2}$. Square of $\frac{\sqrt{2}}{2}$ equals $\frac{1}{2}$.

11. What is the sine of $45^\text{o}$?

    $\frac{\text{sqrt}(2)}{2}$. Standard value for 45° in the unit circle.

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

    1. At 0°, the x-coordinate on the unit circle is 1.

13. State the Pythagorean identity for sine and cosine.

    $\text{sin}^2 \theta + \text{cos}^2 \theta = 1$. Fundamental identity derived from the unit circle.

14. Find the value of $\text{tan}^2 45^\text{o}$.

    1. Square of $\tan 45° = 1$ is 1.

15. State the angle addition formula for sine.

    $\text{sin}(A + B) = \text{sin}A \text{cos}B + \text{cos}A \text{sin}B$. Formula for sine of sum of two angles.

16. Find $\text{cos}(2\theta)$ if $\text{cos}\theta = \frac{1}{2}$ and $\text{sin}\theta = \frac{\text{sqrt}(3)}{2}$.

    $\text{cos}(2\theta) = -\frac{1}{2}$. Using double angle formula with given values.

17. What is the tangent of $60^\text{o}$?

    $\text{tan } 60^\text{o} = \frac{\text{sin } 60^\text{o}}{\text{cos } 60^\text{o}} = \frac{\frac{\text{sqrt}(3)}{2}}{\frac{1}{2}} = \text{sqrt}(3)$. Using the definition $\tan = \frac{\sin}{\cos}$.

18. What is the range of the cosine function?

    [-1, 1]. Cosine values are bounded between -1 and 1.

19. Find $\text{sin}(30^\text{o})$ using cofunction identity.

    $\text{cos}(60^\text{o}) = \frac{1}{2}$. Using $\sin(90° - 60°) = \cos 60°$.

20. What is the tangent of $30^\text{o}$?

    $\frac{1}{\text{sqrt}(3)}$. Using the 30-60-90 triangle ratios.

21. Convert 180 degrees to radians.

    $\text{π}$ radians. Use the conversion factor $\frac{\pi}{180}$.

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

    1. At 90°, the y-coordinate on the unit circle is 1.

23. What is the cosine of $45^\text{o}$?

    $\frac{\text{sqrt}(2)}{2}$. Standard value for 45° in the unit circle.

24. What is the secant of a 60-degree angle?

    2. Secant is the reciprocal of cosine; $\cos 60° = \frac{1}{2}$.

25. What is the sine of a $45^\text{°}$ angle in a right triangle?

    $\frac{\text{√}2}{2}$. In a 45-45-90 triangle, opposite and adjacent sides are equal.

26. What is the sine of $60^\text{o}$?

    $\frac{\sqrt{3}}{2}$. Standard value for 60° in the unit circle.

27. What is the reciprocal of cosine?

    Secant. Secant function is defined as $\sec \theta = \frac{1}{\cos \theta}$.

28. Identify the reciprocal of cosine.

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

29. What is the cosine of $90^\text{o}$?

    0. At 90°, the x-coordinate on the unit circle is 0.

30. State the angle addition formula for cosine.

    $\text{cos}(A + B) = \text{cos}A \text{cos}B - \text{sin}A \text{sin}B$. Formula for cosine of sum of two angles.

31. What is the cosine of $30^\text{o}$?

    $\frac{\text{sqrt}(3)}{2}$. Standard value for 30° in the unit circle.

32. Identify the cofunction identity for cosine.

    $\text{cos}(90^\text{o} - \theta) = \text{sin}(\theta)$. Complementary angle relationship for cosine.

33. State the double angle formula for sine.

    $\sin(2\theta) = 2 \sin(\theta) \cos(\theta)$. Formula for sine of double angle.

34. State the formula for the area of a triangle using sine.

    $\frac{1}{2}ab \sin C$. Uses two sides and the included angle between them.

35. Find the exact value of $\text{sin} 30^\text{o}$.

    $\frac{1}{2}$. In a 30-60-90 triangle, the shortest side is half the hypotenuse.

36. What is the formula for the tangent of an angle in a right triangle?

    $\text{tan} \theta = \frac{\text{opp}}{\text{adj}}$. Tangent equals opposite side divided by adjacent side.

37. What is the value of $\text{sin}^2 45^\text{o}$?

    $\frac{1}{2}$. Square of $\frac{\sqrt{2}}{2}$ equals $\frac{1}{2}$.

38. State the double angle formula for cosine.

    $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$. Formula for cosine of double angle.

39. What is the period of the tangent function?

    $\text{π}$. Tangent completes one cycle every $\pi$ radians.

40. State the Pythagorean identity involving sine and cosine.

    $\text{sin}^2\theta + \text{cos}^2\theta = 1$. Fundamental trigonometric identity derived from the unit circle.

41. Identify the reciprocal function of sine.

    Cosecant (csc). Since $\sin\theta = \frac{1}{\csc\theta}$, cosecant is sine's reciprocal.

42. Identify the cosine of a 0-degree angle.

    1. At 0°, the adjacent side equals the hypotenuse.

43. What is the period of the cosine function?

    $2\text{π}$. Cosine completes one cycle every $2\pi$ radians.

44. State the half angle formula for cosine.

    $\cos(\frac{\theta}{2}) = \sqrt{\frac{1 + \cos(\theta)}{2}}$. Formula for cosine of half angle.

45. What is the sine of a 90-degree angle?

    1. At 90°, the opposite side equals the hypotenuse in a right triangle.

46. What is the period of the sine function?

    $2\text{π}$. Sine completes one cycle every $2\pi$ radians.

47. State the half angle formula for sine.

    $\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{2}}$. Formula for sine of half angle.

48. State the half angle formula for tangent.

    $\text{tan}(\frac{\theta}{2}) = \text{sqrt}(\frac{1 - \text{cos}(\theta)}{1 + \text{cos}(\theta)})$. Formula for tangent of half angle.

49. Find $\cos(2\theta)$ if $\cos \theta = \frac{1}{2}$ and $\sin \theta = \frac{\sqrt{3}}{2}$.

    $\cos(2\theta) = -\frac{1}{2}$. Using double angle formula with given values.

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

    $\frac{1}{2}$. Standard value for 60° in the unit circle.

51. Find the cotangent of a 45-degree angle.

    1. Since $\tan 45° = 1$, cotangent is the reciprocal.

52. State the double angle formula for tangent.

    $\text{tan}(2\theta) = \frac{2\text{tan}(\theta)}{1-\text{tan}^2(\theta)}$. Formula for tangent of double angle.

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

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

54. Find $\sin(2\theta)$ if $\cos\theta = \frac{1}{2}$ and $\sin\theta = \frac{\sqrt{3}}{2}$.

    $\sin(2\theta) = \sqrt{3}$. Using double angle formula with given values.

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