🧭 Level 2: Angle Explorer

Q66. The coordinates of the point on the unit circle corresponding to \( \frac{2\pi}{3} \) radians are:

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

Q67. For the angle \( \frac{5\pi}{6} \), which quadrant does the terminal side lie in, and what is the sign of sine?

(a) Quadrant II, positive sine (b) Quadrant II, negative sine (c) Quadrant III, negative sine

Q68. What is the value of \( \cos(\frac{7\pi}{4}) \) using the unit circle?

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

Q69. If the point \( (-\frac{\sqrt{3}}{2}, -\frac{1}{2}) \) lies on the unit circle, what is the angle in radians?

(a) \( \frac{7\pi}{6} \) (b) \( \frac{5\pi}{6} \) (c) \( \frac{4\pi}{3} \)

Q70. For which angle is tangent undefined on the unit circle?

(a) \( 0 \) (b) \( \frac{\pi}{2} \) (c) \( \pi \)

Q71. Convert \( 330^\circ \) to radians in terms of \( \pi \).

(a) \( \frac{5\pi}{6} \) (b) \( \frac{11\pi}{6} \) (c) \( \frac{7\pi}{4} \)

Q72. Convert \( \frac{5\pi}{8} \) radians to degrees.

(a) \( 112.5^\circ \) (b) \( 120^\circ \) (c) \( 135^\circ \)

Q73. A central angle in a circle of radius 6 cm subtends an arc of length \( 3\pi \) cm. What is the radian measure of this angle?

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

Q74. Find the area of a sector of a circle with radius 4 cm and central angle \( \frac{\pi}{4} \) radians.

(a) \( \pi \) cm\(^2\) (b) \( 2\pi \) cm\(^2\) (c) \( 4\pi \) cm\(^2\)

Q75. If an arc length is 10 cm and the radius of the circle is 5 cm, what is the central angle in degrees?

(a) \( 90^\circ \) (b) \( 114.6^\circ \) (c) \( 120^\circ \)

Q76. In a right-angled triangle ABC, where angle B is \( 90^\circ \), if AB = 8 and AC = 17, find \( \sin(C) \).

(a) \( \frac{8}{17} \) (b) \( \frac{15}{17} \) (c) \( \frac{8}{15} \)

Q77. A ladder 10m long leans against a vertical wall. If the ladder makes an angle of \( 30^\circ \) with the wall, how far is the foot of the ladder from the wall?

(a) \( 5 \) m (b) \( 5\sqrt{3} \) m (c) \( 10 \) m

Q78. From a point on the ground 25m away from the base of a tree, the angle of elevation to the top of the tree is \( 45^\circ \). Find the height of the tree.

(a) \( 25 \) m (b) \( 25\sqrt{3} \) m (c) \( 50 \) m

Q79. A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank is \( 60^\circ \). If the height of the tree is \( 12 \) m, what is the width of the river?

(a) \( 12\sqrt{3} \) m (b) \( \frac{12}{\sqrt{3}} \) m (c) \( 6 \) m

Q80. Simplify the expression: \( \sin^2(\theta) + \cos^2(\theta) + \tan^2(\theta) \).

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

Q81. Simplify: \( \frac{1 - \cos^2(\theta)}{\sin^2(\theta)} \).

(a) 0 (b) 1 (c) \( \tan^2(\theta) \)

Q82. Simplify: \( \sec(\theta) \cdot \cot(\theta) \cdot \sin(\theta) \).

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

Q83. Prove the identity: \( \frac{\sin^2(\theta)}{1 - \cos(\theta)} = 1 + \cos(\theta) \). Which of the following is a step in proving this identity?

(a) Multiply numerator and denominator by \( \sin(\theta) \) (b) Multiply numerator and denominator by \( 1 + \cos(\theta) \) (c) Use the identity \( \tan^2(\theta) + 1 = \sec^2(\theta) \)

Q84. Given \( \sin(\theta) = \frac{4}{5} \) and \( \theta \) is in Quadrant II, find \( \cos(\theta) \).

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

Q85. Given \( \tan(\theta) = -\frac{5}{12} \) and \( \theta \) is in Quadrant IV, find \( \sec(\theta) \).

(a) \( \frac{13}{12} \) (b) \( -\frac{13}{12} \) (c) \( \frac{17}{12} \)

Q86. The graph of \( y = \sin(x) \) has a period of:

(a) \( \pi \) (b) \( 2\pi \) (c) \( 4\pi \)

Q87. What is the amplitude of the function \( y = 2\cos(x) \)?

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

Q88. The maximum value of \( y = \cos(x) \) is:

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

Q89. The graph of \( y = \sin(x) \) passes through which point?

(a) \( (\pi, 1) \) (b) \( (\frac{\pi}{2}, 0) \) (c) \( (\frac{\pi}{2}, 1) \)

Q90. Which function has a graph that is symmetric about the y-axis?

(a) \( y = \sin(x) \) (b) \( y = \tan(x) \) (c) \( y = \cos(x) \)

Q91. The coordinates of the point on the unit circle corresponding to \( \frac{3\pi}{4} \) radians are:

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

Q92. For the angle \( \frac{4\pi}{3} \), which quadrant does the terminal side lie in, and what is the sign of cosine?

(a) Quadrant III, positive cosine (b) Quadrant III, negative cosine (c) Quadrant IV, negative cosine

Q93. What is the value of \( \sin(\frac{11\pi}{6}) \) using the unit circle?

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

Q94. If the point \( (\frac{1}{2}, -\frac{\sqrt{3}}{2}) \) lies on the unit circle, what is the angle in radians?

(a) \( \frac{5\pi}{3} \) (b) \( \frac{7\pi}{6} \) (c) \( \frac{11\pi}{6} \)

Q95. For which angle is cotangent undefined on the unit circle?

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

Q96. Convert \( 240^\circ \) to radians in terms of \( \pi \).

(a) \( \frac{4\pi}{3} \) (b) \( \frac{5\pi}{4} \) (c) \( \frac{7\pi}{6} \)

Q97. Convert \( \frac{7\pi}{9} \) radians to degrees.

(a) \( 140^\circ \) (b) \( 150^\circ \) (c) \( 160^\circ \)

Q98. A central angle of \( 2 \) radians in a circle of radius 3 cm subtends an arc of length:

(a) 1.5 cm (b) 5 cm (c) 6 cm

Q99. Find the area of a sector of a circle with radius 6 cm and arc length \( 2\pi \) cm.

(a) \( 6\pi \) cm\(^2\) (b) \( 12\pi \) cm\(^2\) (c) \( 18\pi \) cm\(^2\)

Q100. If the area of a sector is \( 9\pi \) cm\(^2\) and the radius is 3 cm, what is the central angle in radians?

(a) \( 2\pi \) (b) \( 3\pi \) (c) \( 6\pi \)

Q101. In a right-angled triangle PQR, where angle Q is \( 90^\circ \), if PQ = 5 and PR = 13, find \( \cos(P) \).

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

Q102. A ramp is 15m long and rises to a height of 3m. What angle does the ramp make with the ground? (Use \( \sin^{-1}(0.2) \approx 11.5^\circ \))

(a) \( 11.5^\circ \) (b) \( 30^\circ \) (c) \( 45^\circ \)

Q103. From a window 15m high above the ground in a building, the angle of depression to a point on the ground is \( 60^\circ \). How far is the point from the base of the building?

(a) \( 15\sqrt{3} \) m (b) \( \frac{15}{\sqrt{3}} \) m (c) \( 30 \) m

Q104. Simplify the expression: \( \frac{\cos(\theta)}{\sec(\theta)} + \sin^2(\theta) \).

(a) 0 (b) 1 (c) \( \cos^2(\theta) \)

Q105. Simplify: \( \csc^2(\theta) - \cot^2(\theta) + \sin^2(\theta) \).

(a) \( \cos^2(\theta) \) (b) \( \sin^2(\theta) \) (c) \( 1 + \sin^2(\theta) \)

Q106. Simplify: \( \frac{1}{\tan(\theta)} \cdot \frac{1}{\cot(\theta)} \).

(a) 0 (b) 1 (c) \( \tan^2(\theta) \)

Q107. Prove the identity: \( \frac{\cos(\theta)}{1 + \sin(\theta)} + \frac{\cos(\theta)}{1 - \sin(\theta)} = 2\sec(\theta) \). Which identity is most directly used in simplifying the left side?

(a) Pythagorean Identity (b) Quotient Identity (c) Reciprocal Identity

Q108. Given \( \cos(\theta) = -\frac{12}{13} \) and \( \theta \) is in Quadrant III, find \( \sin(\theta) \).

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

Q109. Given \( \cot(\theta) = \frac{3}{4} \) and \( \theta \) is in Quadrant I, find \( \csc(\theta) \).

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

Q110. The graph of \( y = \cos(x) \) has a period of:

(a) \( \pi \) (b) \( 2\pi \) (c) \( 4\pi \)

Q111. What is the amplitude of the function \( y = 0.5\sin(x) \)?

(a) 0.5 (b) 1 (c) 2

Q112. The minimum value of \( y = \sin(x) \) is:

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

Q113. The graph of \( y = \cos(x) \) passes through which point?

(a) \( (\pi, 0) \) (b) \( (\frac{\pi}{2}, 1) \) (c) \( (\frac{\pi}{2}, 0) \)

Q114. Which function has a graph that is symmetric about the origin?

(a) \( y = \cos(x) \) (b) \( y = \sin(x) \) (c) \( y = \) constant

Q115. What is the period of the function \( y = \sin(2x) \)?

(a) \( \pi \) (b) \( 2\pi \) (c) \( 4\pi \)

Q116. What is the phase shift of the function \( y = \cos(x - \frac{\pi}{2}) \)?

(a) \( \frac{\pi}{2} \) to the left (b) \( \frac{\pi}{2} \) to the right (c) \( \pi \) to the right

Q117. What is the amplitude of the function \( y = -3\sin(x) \)?

(a) -3 (b) 3 (c) 1

Q118. The range of the function \( y = \cos(x) \) is:

(a) \( (-\infty, \infty) \) (b) \( [-1, 1] \) (c) \( [0, 1] \)

Q119. The graph of \( y = \sin(x) \) is increasing on the interval:

(a) \( (\pi, 2\pi) \) (b) \( (0, \pi) \) (c) \( (-\frac{\pi}{2}, \frac{\pi}{2}) \)

Q120. What is the domain of the function \( y = \sin(x) \)?

(a) \( [-1, 1] \) (b) \( [0, 2\pi] \) (c) \( (-\infty, \infty) \)