Trigonometry - ACT Math

1. What is $\cos\left(\frac{\pi}{3}\right)$?

   $\frac{1}{2}$. From 60° angle in 30-60-90 triangle.

2. What is the angle of depression if the adjacent side is 1 and the opposite side is 1?

   45 degrees. Equal opposite and adjacent sides create a 45° angle.

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

   $2$. Secant is $\frac{1}{\cos(60^\circ)} = \frac{1}{1/2}$

4. What is the cotangent of an angle in terms of tangent?

   $\text{cot} \theta = \frac{1}{\text{tan} \theta}$. Cotangent is the reciprocal of tangent.

5. What are the side ratios for a $45^\circ$-$45^\circ$-$90^\circ$ triangle (legs to hypotenuse)?

   $1:1:\sqrt{2}$. Isosceles right triangle with legs of 1 and hypotenuse $\sqrt{2}$.

6. What is the period of $y=\tan(Bx)$ in terms of $B$?

   $\frac{\pi}{|B|}$. Tangent has period $\pi$, so with coefficient $B$ it's $\frac{\pi}{|B|}$.

7. What is the cosine of a 0-degree angle?

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

8. Find the value of $\text{tan}^{-1}(1)$ in degrees.

   $45^\text{o}$. Inverse tangent of 1 gives the angle where $\tan(\theta) = 1$.

9. What is $\sin(2\pi)$ on the unit circle?

   $0$. At $2\pi$, back to (1,0), so sine is 0.

10. What is the cosecant of a 30-degree angle?

    $2$. Cosecant is $\frac{1}{\sin(30°)} = \frac{1}{1/2}$.

11. What is the sine of a $30^\text{o}$ angle?

    $\frac{1}{2}$. This is a special angle value from the 30-60-90 triangle.

12. What is the range of the cosine function?

    [-1, 1]. Cosine oscillates between -1 and 1 for all angles.

13. What is the radian-to-degree conversion formula for an angle $\theta$ in radians?

    $\theta_{\circ}=\theta_{\text{rad}}\cdot\frac{180}{\pi}$. Multiply radians by $\frac{180}{\pi}$ to convert to degrees.

14. What is the secant of an angle in terms of cosine?

    $\text{sec} \theta = \frac{1}{\text{cos} \theta}$. Secant is the reciprocal of cosine.

15. What is the reciprocal of tangent?

    Cotangent. Reciprocal function: $\cot(\theta) = \frac{1}{\tan(\theta)}$.

16. What is the arc length formula for a circle with radius $r$ and central angle $\theta$ (in radians)?

    $s=r\theta$. Arc length equals radius times central angle in radians.

17. What is the cosecant of an angle in terms of sine?

    $\text{csc} \theta = \frac{1}{\text{sin} \theta}$. Cosecant is the reciprocal of sine.

18. What is the cosine of a $90^\text{o}$ angle?

    0. At 90°, the adjacent side is zero.

19. What is the value of $\text{tan}(0^\text{o})$?

    0. At 0°, opposite side has zero length.

20. State the Pythagorean identity for sine and cosine.

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

21. What is $\tan\left(\frac{\pi}{6}\right)$?

    $\frac{\sqrt{3}}{3}$. $\frac{\sin(\pi/6)}{\cos(\pi/6)} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}}$.

22. Find the angle if the tangent is 1. Provide the acute angle.

    45 degrees. Use inverse tangent to find the reference angle.

23. What is the sector area formula for radius $r$ and central angle $\theta$ (in radians)?

    $A=\frac{1}{2}r^2\theta$. Sector area is half the product of radius squared and angle.

24. What is the angle of elevation if the opposite side is 1 and the adjacent side is $\sqrt{3}$?

    30 degrees. Use $\tan^{-1}(\frac{1}{\sqrt{3}})$ to find the angle.

25. Convert $\frac{\text{π}}{2}$ radians to degrees.

    90 degrees. $\frac{\pi}{2}$ radians equals 90° by definition.

26. What is the period of $y = \cos(2x)$?

    $\pi$. Period formula: $\frac{2\pi}{|b|}$ where $b = 2$.

27. What is the cosine of an angle whose cosecant is 2?

    $\frac{\sqrt{3}}{2}$. If $\csc = 2$, then $\sin = \frac{1}{2}$, use Pythagorean identity.

28. What is $\sin\left(\frac{\pi}{4}\right)$?

    $\frac{\sqrt{2}}{2}$. From 45° angle in 45-45-90 triangle.

29. What is the tangent of a 45-degree angle?

    1. 45° creates an isosceles right triangle.

30. Convert $\frac{\text{π}}{3}$ radians to degrees.

    60 degrees. Multiply radians by $\frac{180}{\pi}$.

31. State the Pythagorean identity for tangent and secant.

    $1 + \text{tan}^2 \theta = \text{sec}^2 \theta$. Derived from the Pythagorean identity and reciprocals.

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

    0. Since $\sin(0°) = 0$ and $\cos(0°) = 1$.

33. Find the angle if the sine is $\frac{1}{2}$. Provide the acute angle.

    30 degrees. Use inverse sine to find the reference angle.

34. State the formula for the tangent of an angle.

    $\text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta}$. Tangent equals the ratio of sine to cosine.

35. What is the degree-to-radian conversion formula for an angle $\theta$ in degrees?

    $\theta_{\text{rad}}=\theta_{\circ}\cdot\frac{\pi}{180}$. Multiply degrees by $\frac{\pi}{180}$ to convert to radians.

36. What is $\sin(0)$ on the unit circle?

    $0$. At angle 0, the point is at (1,0), so sine is 0.

37. Find the angle if the cosine is $\frac{1}{2}$. Provide the acute angle.

    60 degrees. Use inverse cosine to find the reference angle.

38. What is $\tan\left(\frac{\pi}{3}\right)$?

    $\sqrt{3}$. $\frac{\sin(\pi/3)}{\cos(\pi/3)} = \frac{\sqrt{3}/2}{1/2} = \sqrt{3}$.

39. What is the tangent of a $45^\text{o}$ angle?

    1. In a 45-45-90 triangle, opposite and adjacent sides are equal.

40. What is $\cos(2\pi)$ on the unit circle?

    $1$. At $2\pi$, back to (1,0), so cosine is 1.

41. What is the cosine of a $60^\text{o}$ angle?

    $\frac{1}{2}$. From the 30-60-90 triangle, $\cos(60°) = \sin(30°)$.

42. What is the cosine of a 60-degree angle?

    $\frac{1}{2}$. 60° is a special angle in the unit circle.

43. Convert 180 degrees to radians.

    $\pi$. 180° equals $\pi$ radians by definition.

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

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

45. What is the angle of elevation if the opposite side is 1 and the adjacent side is $\text{√3}$?

    30 degrees. Use $\tan^{-1}(\frac{1}{\sqrt{3}})$ to find the angle.

46. What is the secant of a 90-degree angle?

    Undefined. Division by zero makes secant undefined.

47. For which angle is $\text{sin} \theta = 1$?

    $90^\text{o}$. Sine reaches its maximum value of 1 at 90°.

48. What is the sine of a 30-degree angle?

    $\frac{1}{2}$. 30° is a special angle in the unit circle.

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

    0. At 0°, there's no vertical component.

50. What is the sine of $180^\text{o}$?

    0. At 180°, we're on the negative x-axis where y = 0.

51. What is the sine of $0^\text{o}$?

    0. At 0°, the opposite side is zero.

52. What is the cosecant of a 0-degree angle?

    Undefined. Division by zero makes cosecant undefined.

53. What are the side ratios for a $30^\circ$-$60^\circ$-$90^\circ$ triangle (short:long:hyp)?

    $1:\sqrt{3}:2$. Half of an equilateral triangle: short leg, long leg, hypotenuse.

54. What is $\cos(\pi)$ on the unit circle?

    $-1$. At $\pi$, the point is at (-1,0), so cosine is -1.

55. What is the cotangent of a 0-degree angle?

    Undefined. Division by zero makes cotangent undefined.

56. What is the sine of a $90^\text{o}$ angle?

    1. At 90°, the opposite side equals the hypotenuse.

57. What is the period of $y=\sin(Bx)$ or $y=\cos(Bx)$ in terms of $B$?

    $\frac{2\pi}{|B|}$. Time for one complete cycle is $\frac{2\pi}{|B|}$.

58. What is the formula for the distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ in the plane?

    $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Pythagorean theorem applied to coordinate differences.

59. What is the tangent of $180^\text{o}$?

    0. Since $\sin(180°) = 0$ and $\cos(180°) = -1$.

60. What is the cotangent of a 45-degree angle?

    1. At 45°, sine equals cosine, so their ratio is 1.

61. What is $\tan\left(\frac{\pi}{4}\right)$?

    $1$. $\frac{\sin(\pi/4)}{\cos(\pi/4)} = \frac{\sqrt{2}/2}{\sqrt{2}/2} = 1$.

62. What is the value of $\text{cos}(90^\text{o})$?

    0. At 90°, there's no horizontal component.

63. What are the right-triangle definitions of $\sin(\theta)$, $\cos(\theta)$, and $\tan(\theta)$?

    $\sin=\frac{\text{opp}}{\text{hyp}},\ \cos=\frac{\text{adj}}{\text{hyp}},\ \tan=\frac{\text{opp}}{\text{adj}}$. SOH-CAH-TOA: ratios of opposite, adjacent sides to hypotenuse.

64. What is $\sin(\pi)$ on the unit circle?

    $0$. At $\pi$, the point is at (-1,0), so sine is 0.

65. Find the value of $\text{cos}^{-1}(0)$ in degrees.

    $90^\text{o}$. Inverse cosine of 0 gives the angle where $\cos(\theta) = 0$.

66. Find the exact value of $\sin(\frac{\pi}{6})$.

    $\frac{1}{2}$. This is $\sin(30^\circ)$, a special angle value.

67. State the reciprocal identity for tangent.

    $\tan \theta = \frac{1}{\cot \theta}$. Tangent and cotangent are reciprocals.

68. What is the Pythagorean identity relating $\sin(\theta)$ and $\cos(\theta)$?

    $\sin^2(\theta)+\cos^2(\theta)=1$. From the Pythagorean theorem applied to the unit circle.

69. State the Pythagorean identity involving sine and cosine.

    $\text{sin}^2 \theta + \text{cos}^2 \theta = 1$. This fundamental identity comes from the Pythagorean theorem.

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

    1. At 90°, the point is at the top of the unit circle.

71. What is $\sin\left(\frac{\pi}{6}\right)$?

    $\frac{1}{2}$. From 30° angle in 30-60-90 triangle.

72. State the formula for converting degrees to radians.

    $\theta \text{ (radians)} = \theta \text{ (degrees)} \times \frac{\pi}{180}$. Multiply by $\frac{\pi}{180}$ to convert degrees to radians.

73. What is $\cos\left(\frac{3\pi}{2}\right)$ on the unit circle?

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

74. State the co-function identity for sine and cosine.

    $\text{sin}(\theta) = \text{cos}(90^\text{o} - \theta)$. Complementary angles have swapped sine and cosine.

75. What are the reciprocal identities for $\csc(\theta)$, $\sec(\theta)$, and $\cot(\theta)$?

    $\csc=\frac{1}{\sin},\ \sec=\frac{1}{\cos},\ \cot=\frac{1}{\tan}$. Each function equals 1 divided by its corresponding primary function.

76. State the formula for $\text{tan}(\theta)$ in terms of sine and cosine.

    $\text{tan}(\theta) = \frac{\text{sin}(\theta)}{\text{cos}(\theta)}$. Definition of tangent as the ratio of sine to cosine.

77. What is $\cos\left(\frac{\pi}{4}\right)$?

    $\frac{\sqrt{2}}{2}$. From 45° angle in 45-45-90 triangle.

78. What are the quotient identities for $\tan(\theta)$ and $\cot(\theta)$ in terms of $\sin$ and $\cos$?

    $\tan=\frac{\sin}{\cos},\ \cot=\frac{\cos}{\sin}$. Tangent and cotangent are ratios of sine and cosine.

79. What is $\cos\left(\frac{\pi}{6}\right)$?

    $\frac{\sqrt{3}}{2}$. From 30° angle in 30-60-90 triangle.

80. What is the cosine of $180^\text{o}$?

    -1. At 180°, we're on the negative x-axis where x = -1.

81. State the double angle formula for cosine.

    $\cos(2\theta) = \cos^2 \theta - \sin^2 \theta$. One of the three forms of the double angle formula.

82. What is the sine of an angle whose secant is 2?

    $\frac{\sqrt{3}}{2}$. If $\sec = 2$, then $\cos = \frac{1}{2}$, use Pythagorean identity.

83. What is the tangent of a $90^\text{o}$ angle?

    Undefined. Division by zero occurs since $\cos(90°) = 0$.

84. State the double angle formula for sine.

    $\text{sin}(2\theta) = 2\text{sin}(\theta)\text{cos}(\theta)$. Double angle formula using product-to-sum identity.

85. What is the reciprocal of sine?

    Cosecant. Reciprocal function: $\csc(\theta) = \frac{1}{\sin(\theta)}$.

86. What is the tangent of a 90-degree angle?

    Undefined. Division by zero makes tangent undefined.

87. Convert 45 degrees to radians.

    $\frac{\text{π}}{4}$. Multiply degrees by $\frac{\pi}{180}$.

88. State the Pythagorean identity for cotangent and cosecant.

    $1 + \text{cot}^2 \theta = \text{csc}^2 \theta$. Derived from the Pythagorean identity and reciprocals.

89. What is $\sin\left(\frac{\pi}{2}\right)$ on the unit circle?

    $1$. At $\frac{\pi}{2}$, the point is at (0,1), so sine is 1.

90. Identify the phase shift in $y = \text{sin}(x - \frac{\text{\textpi}}{3})$.

    $\frac{\text{\textpi}}{3}$ to the right. Subtracting from $x$ shifts the graph to the right.

91. What is the period of $y = \tan(x)$?

    $\pi$. Tangent has period $\pi$, half that of sine and cosine.

92. What is $\sin\left(\frac{3\pi}{2}\right)$ on the unit circle?

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

93. What is the amplitude of $y=A\sin(Bx)$ or $y=A\cos(Bx)$ in terms of $A$?

    $|A|$. Maximum distance from equilibrium position is $|A|$.

94. What is the formula for the area of a triangle using sine?

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

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

    All points $(x,y)$ with $x^2+y^2=1$. Circle centered at origin with radius 1.

96. What is $\sin\left(\frac{\pi}{3}\right)$?

    $\frac{\sqrt{3}}{2}$. From 60° angle in 30-60-90 triangle.

97. What are the coordinates on the unit circle corresponding to angle $\theta$?

    $(\cos(\theta),\sin(\theta))$. Cosine gives x-coordinate, sine gives y-coordinate.

98. What is the range of the sine function?

    [-1, 1]. Sine oscillates between -1 and 1 for all angles.

99. What is $\cos(0)$ on the unit circle?

    $1$. At angle 0, the point is at (1,0), so cosine is 1.

100. What is the reciprocal of cosine?

     Secant. Reciprocal function: $\sec(\theta) = \frac{1}{\cos(\theta)}$.

101. What is $\cos\left(\frac{\pi}{2}\right)$ on the unit circle?

     $0$. At $\frac{\pi}{2}$, the point is at (0,1), so cosine is 0.

102. Identify the amplitude in $y = 3\text{sin}(x)$.

     3. The coefficient of the sine function determines amplitude.

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