🚀 Level 1: Trigonometry Launchpad

Q1. An angle is said to be in standard position if its vertex is at the:

(a) origin (b) first quadrant (c) x-axis

Q2. Which of the following is coterminal with an angle of \(60^\circ\)?

(a) \(-60^\circ\) (b) \(420^\circ\) (c) \(120^\circ\)

Q3. Convert \( \frac{\pi}{3} \) radians to degrees.

(a) \(30^\circ\) (b) \(45^\circ\) (c) \(60^\circ\)

Q4. An angle of \( 270^\circ \) is a:

(a) acute angle (b) obtuse angle (c) reflex angle

Q5. The angle \( -45^\circ \) is in which quadrant when in standard position?

(a) Quadrant I (b) Quadrant II (c) Quadrant IV

Q6. What is the radian measure of a right angle?

(a) \( \frac{\pi}{4} \) (b) \( \frac{\pi}{3} \) (c) \( \frac{\pi}{2} \)

Q7. If an angle is \(135^\circ\), in which quadrant does its terminal side lie?

(a) Quadrant I (b) Quadrant II (c) Quadrant III

Q8. Which angle is NOT coterminal with \(30^\circ\)?

(a) \(390^\circ\) (b) \(-330^\circ\) (c) \(300^\circ\)

Q9. Convert \( \frac{2\pi}{5} \) radians to degrees.

(a) \(72^\circ\) (b) \(108^\circ\) (c) \(120^\circ\)

Q10. An angle of \( \frac{7\pi}{6} \) radians is in which quadrant?

(a) Quadrant II (b) Quadrant III (c) Quadrant IV

Q11. If \( \sin(\theta) = \frac{3}{5} \) in a right-angled triangle, what is \( \cos(\theta) \)?

(a) \( \frac{4}{5} \) (b) \( \frac{3}{4} \) (c) \( \frac{5}{4} \)

Q12. In a right-angled triangle, if the opposite side is 5 and the adjacent side is 12, what is \( \tan(\theta) \)?

(a) \( \frac{12}{5} \) (b) \( \frac{5}{13} \) (c) \( \frac{5}{12} \)

Q13. What is the value of \( \sin(30^\circ) \)?

(a) \( \frac{1}{2} \) (b) \( \frac{\sqrt{3}}{2} \) (c) \( \frac{\sqrt{2}}{2} \)

Q14. Find the value of \( \cos(45^\circ) \).

(a) \( \frac{1}{2} \) (b) \( \frac{\sqrt{3}}{2} \) (c) \( \frac{\sqrt{2}}{2} \)

Q15. What is the value of \( \tan(60^\circ) \)?

(a) 1 (b) \( \frac{1}{\sqrt{3}} \) (c) \( \sqrt{3} \)

Q16. In a 30-60-90 triangle, the ratio of sides opposite to 30°, 60°, and 90° angles is:

(a) 1 : 1 : \( \sqrt{2} \) (b) 1 : \( \sqrt{3} \) : 2 (c) \( \sqrt{2} \) : \( \sqrt{2} \) : 2

Q17. What is the value of \( \sec(0^\circ) \)?

(a) 0 (b) 1 (c) undefined

Q18. Find the value of \( \csc(90^\circ) \).

(a) 0 (b) 1 (c) undefined

Q19. What is the value of \( \cot(45^\circ) \)?

(a) 0 (b) 1 (c) undefined

Q20. Which reciprocal function is the inverse of sine?

(a) secant (b) cosecant (c) cotangent

Q21. If \( \sin(\theta) = \frac{1}{\csc(\theta)} \), this is a(n):

(a) equation (b) identity (c) inequality

Q22. Express \( \tan(\theta) \) in terms of \( \sin(\theta) \) and \( \cos(\theta) \).

(a) \( \sin(\theta) \cdot \cos(\theta) \) (b) \( \frac{\cos(\theta)}{\sin(\theta)} \) (c) \( \frac{\sin(\theta)}{\cos(\theta)} \)

Q23. If \( \cos(\theta) = \frac{x}{r} \) and \( \sin(\theta) = \frac{y}{r} \), what does \( r \) represent in the unit circle context?

(a) radius (b) hypotenuse (c) both radius and hypotenuse

Q24. In a right triangle, if hypotenuse is 10 and opposite side is 6, what is the adjacent side using Pythagorean Theorem?

(a) 8 (b) 16 (c) \( \sqrt{136} \)

Q25. The Pythagorean Identity states:

(a) \( \sin^2(\theta) + \cos^2(\theta) = 1 \) (b) \( \tan^2(\theta) + \sec^2(\theta) = 1 \) (c) \( \cot^2(\theta) - \csc^2(\theta) = 1 \)

Q26. Simplify: \( \frac{\sin(\theta)}{\csc(\theta)} \).

(a) \( \sin^2(\theta) \) (b) \( \cos^2(\theta) \) (c) \( \tan^2(\theta) \)

Q27. Which of the following is a form of Pythagorean Identity?

(a) \( 1 + \tan^2(\theta) = \csc^2(\theta) \) (b) \( 1 + \cot^2(\theta) = \csc^2(\theta) \) (c) \( \sec^2(\theta) - \tan^2(\theta) = -1 \)

Q28. A ladder leans against a wall making an angle of \(60^\circ\) with the ground. If the foot of the ladder is 2m away from the wall, what is the height of the ladder on the wall?

(a) \( 2\sqrt{3} \) m (b) 4 m (c) 1 m

Q29. From the top of a cliff 20m high, the angle of depression to a boat is \(30^\circ\). How far is the boat from the base of the cliff?

(a) \( 20\sqrt{3} \) m (b) 40 m (c) 10 m

Q30. A tower casts a shadow of 30m when the angle of elevation of the sun is \(45^\circ\). What is the height of the tower?

(a) 15 m (b) 30 m (c) 60 m

Q31. If \( \sin(\theta) = \frac{\sqrt{3}}{2} \) and \( \theta \) is in Quadrant II, what is \( \theta \) in degrees?

(a) \(60^\circ\) (b) \(120^\circ\) (c) \(300^\circ\)

Q32. What is \( \cos(180^\circ) \)?

(a) -1 (b) 0 (c) 1

Q33. What is the value of \( \tan(\pi) \)?

(a) 0 (b) 1 (c) undefined

Q34. The coordinates of a point on the unit circle corresponding to angle \( \pi \) are:

(a) (0, 1) (b) (1, 0) (c) (-1, 0)

Q35. Which quadrant has sine positive and cosine negative?

(a) Quadrant I (b) Quadrant II (c) Quadrant III

Q36. What is the reference angle for \( 150^\circ \)?

(a) \(30^\circ\) (b) \(60^\circ\) (c) \(150^\circ\)

Q37. What is the reference angle for \( \frac{5\pi}{4} \) radians?

(a) \( \frac{\pi}{4} \) (b) \( \frac{\pi}{3} \) (c) \( \frac{3\pi}{4} \)

Q38. In which quadrant is \( \theta \) if \( \sin(\theta) < 0 \) and \( \cos(\theta) > 0 \)?

(a) Quadrant II (b) Quadrant III (c) Quadrant IV

Q39. What is the sign of tangent in Quadrant III?

(a) positive (b) negative (c) zero

Q40. If \( \tan(\theta) = -1 \) and \( \theta \) is in Quadrant II, what is \( \theta \) in radians?

(a) \( \frac{3\pi}{4} \) (b) \( \frac{5\pi}{4} \) (c) \( \frac{7\pi}{4} \)

Q41. What is the value of \( \sin(90^\circ) \)?

(a) 0 (b) 1 (c) -1

Q42. Find the value of \( \cos(0) \).

(a) 0 (b) 1 (c) -1

Q43. What is the value of \( \tan(270^\circ) \)?

(a) 0 (b) 1 (c) undefined

Q44. The coordinates of a point on the unit circle for \( \frac{3\pi}{2} \) are:

(a) (0, -1) (b) (-1, 0) (c) (1, 0)

Q45. In which quadrant are both sine and cosine negative?

(a) Quadrant II (b) Quadrant III (c) Quadrant IV

Q46. What is the reference angle for \( 225^\circ \)?

(a) \(30^\circ\) (b) \(45^\circ\) (c) \(60^\circ\)

Q47. What is the reference angle for \( \frac{7\pi}{6} \) radians?

(a) \( \frac{\pi}{6} \) (b) \( \frac{\pi}{4} \) (c) \( \frac{\pi}{3} \)

Q48. In which quadrant is \( \theta \) if \( \sin(\theta) > 0 \) and \( \tan(\theta) < 0 \)?

(a) Quadrant I (b) Quadrant II (c) Quadrant III

Q49. What is the sign of cosine in Quadrant IV?

(a) positive (b) negative (c) zero

Q50. If \( \cot(\theta) = 1 \) and \( \theta \) is in Quadrant III, what is \( \theta \) in degrees?

(a) \(135^\circ\) (b) \(225^\circ\) (c) \(315^\circ\)

Q51. What is \( \sin(0^\circ) \)?

(a) 0 (b) 1 (c) -1

Q52. Find the value of \( \cos(90^\circ) \).

(a) 0 (b) 1 (c) -1

Q53. What is the value of \( \tan(180^\circ) \)?

(a) 0 (b) 1 (c) undefined

Q54. The coordinates of a point on the unit circle for \( 0 \) radians are:

(a) (1, 0) (b) (0, 1) (c) (-1, 0)

Q55. In which quadrant is cosine positive and sine negative?

(a) Quadrant I (b) Quadrant IV (c) Quadrant II

Q56. What is the reference angle for \( 300^\circ \)?

(a) \(30^\circ\) (b) \(45^\circ\) (c) \(60^\circ\)

Q57. What is the reference angle for \( \frac{11\pi}{6} \) radians?

(a) \( \frac{\pi}{6} \) (b) \( \frac{\pi}{4} \) (c) \( \frac{\pi}{3} \)

Q58. In which quadrant is \( \theta \) if \( \cos(\theta) < 0 \) and \( \tan(\theta) > 0 \)?

(a) Quadrant II (b) Quadrant III (c) Quadrant IV

Q59. What is the sign of sine in Quadrant III?

(a) positive (b) negative (c) zero

Q60. If \( \sin(\theta) = -\frac{\sqrt{2}}{2} \) and \( \theta \) is in Quadrant IV, what is \( \theta \) in degrees?

(a) \(225^\circ\) (b) \(315^\circ\) (c) \(405^\circ\)

Q61. What is the approximate height of a building that casts a 40m shadow when the angle of elevation of the sun is \(30^\circ\)? (Use \( \sqrt{3} \approx 1.732 \))

(a) 20 m (b) 23.09 m (c) 34.64 m

Q62. From a point 10m away from a tree, the angle of elevation to the top of the tree is \(60^\circ\). Find the height of the tree.

(a) \(10\sqrt{3}\) m (b) 20 m (c) 30 m

Q63. If \( \tan(\theta) = \frac{4}{3} \), find \( \cot(\theta) \).

(a) \( \frac{3}{4} \) (b) \( \frac{5}{3} \) (c) \( \frac{4}{5} \)

Q64. If \( \csc(\theta) = 2 \), find \( \sin(\theta) \).

(a) 1 (b) \( \frac{1}{2} \) (c) 2

Q65. Simplify the expression: \( \cos(\theta) \cdot \sec(\theta) \).

(a) 0 (b) 1 (c) \( \tan(\theta) \)