🧭 Level 3: Trigonometry Navigator

Q121. Solve the equation \( 2\sin(x) - 1 = 0 \) for \( x \) in the interval \( [0, 2\pi) \).

(a) \( x = \frac{\pi}{6}, \frac{5\pi}{6} \) (b) \( x = \frac{\pi}{3}, \frac{2\pi}{3} \) (c) \( x = \frac{\pi}{4}, \frac{3\pi}{4} \)

Q122. Find the general solution for \( \cos(x) = \frac{\sqrt{3}}{2} \).

(a) \( x = \frac{\pi}{6} + 2n\pi \) and \( x = \frac{11\pi}{6} + 2n\pi \) (b) \( x = \pm \frac{\pi}{6} + 2n\pi \) (c) \( x = \frac{\pi}{3} + 2n\pi \) and \( x = \frac{5\pi}{3} + 2n\pi \)

Q123. Solve for \( \tan(x) = -1 \) in the interval \( [0, \pi) \).

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

Q124. Find all solutions for \( 2\sin^2(x) - \sin(x) - 1 = 0 \) in \( [0, 2\pi) \).

(a) \( x = \frac{\pi}{2}, \frac{7\pi}{6}, \frac{11\pi}{6} \) (b) \( x = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{3\pi}{2} \) (c) \( x = \frac{\pi}{2}, \frac{5\pi}{6}, \frac{7\pi}{6} \)

Q125. Solve \( \cos(2x) = \frac{1}{2} \) for \( x \) in \( [0, \pi) \).

(a) \( x = \frac{\pi}{6}, \frac{5\pi}{6} \) (b) \( x = \frac{\pi}{3}, \frac{2\pi}{3} \) (c) \( x = \frac{\pi}{6}, \frac{11\pi}{6} \)

Q126. Find the solutions to \( \sin(3x) = 0 \) in \( [0, 2\pi) \).

(a) \( x = 0, \frac{\pi}{3}, \frac{2\pi}{3}, \pi, \frac{4\pi}{3}, \frac{5\pi}{3} \) (b) \( x = 0, \frac{\pi}{3}, \frac{2\pi}{3}, \pi \) (c) \( x = \frac{\pi}{3}, \frac{2\pi}{3}, \pi, \frac{4\pi}{3}, \frac{5\pi}{3}, 2\pi \)

Q127. Solve \( 2\cos^2(x) - 3\cos(x) + 1 = 0 \) for \( x \) in \( [0, 2\pi) \).

(a) \( x = 0, \frac{\pi}{3}, \frac{5\pi}{3} \) (b) \( x = 0, \frac{\pi}{2}, \frac{3\pi}{2} \) (c) \( x = \frac{\pi}{3}, \frac{\pi}{2}, \frac{5\pi}{3} \)

Q128. Find the solution(s) to \( \sin(x) = \cos(x) \) in \( [0, 2\pi) \).

(a) \( x = \frac{\pi}{4}, \frac{5\pi}{4} \) (b) \( x = \frac{\pi}{4}, \frac{3\pi}{4} \) (c) \( x = \frac{3\pi}{4}, \frac{5\pi}{4} \)

Q129. Solve \( \tan^2(x) = 1 \) for \( x \) in \( [0, 2\pi) \).

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

Q130. Find the general solution of \( \sin(x) = -\frac{1}{2} \).

(a) \( x = \frac{7\pi}{6} + 2n\pi \) and \( x = \frac{11\pi}{6} + 2n\pi \) (b) \( x = \frac{7\pi}{6} + n\pi \) (c) \( x = \frac{11\pi}{6} + 2n\pi \)

Q131. Describe the transformation from \( y = \cos(x) \) to \( y = \cos(x) + 2 \).

(a) Shifted 2 units to the right (b) Shifted 2 units upward (c) Stretched vertically by a factor of 2

Q132. Identify the amplitude of \( y = -3\sin(x) \).

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

Q133. Determine the period of \( y = \sin(2x) \).

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

Q134. Find the phase shift of \( y = \cos(x - \frac{\pi}{3}) \).

(a) \( \frac{\pi}{3} \) to the left (b) \( \frac{\pi}{3} \) to the right (c) \( -\frac{\pi}{3} \) to the right

Q135. Identify the vertical shift of \( y = 2\sin(x) - 1 \).

(a) 1 unit upward (b) 1 unit downward (c) 2 units upward

Q136. What is the value of \( \arcsin(\frac{\sqrt{3}}{2}) \) in radians?

(a) \( \frac{\pi}{6} \) (b) \( \frac{\pi}{3} \) (c) \( \frac{\pi}{2} \)

Q137. Evaluate \( \arccos(-\frac{1}{2}) \) in radians.

(a) \( \frac{\pi}{3} \) (b) \( \frac{2\pi}{3} \) (c) \( \frac{5\pi}{6} \)

Q138. What is the domain of \( y = \arcsin(x) \)?

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

Q139. Determine the range of \( y = \arccos(x) \).

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

Q140. Evaluate \( \arctan(-1) \) in radians.

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

Q141. Use the sum formula to find the exact value of \( \cos(75^\circ) \).

(a) \( \frac{\sqrt{6} - \sqrt{2}}{4} \) (b) \( \frac{\sqrt{6} + \sqrt{2}}{4} \) (c) \( \frac{\sqrt{2} - \sqrt{6}}{4} \)

Q142. Apply the difference formula to simplify \( \sin(x - \frac{\pi}{2}) \).

(a) \( \cos(x) \) (b) \( -\cos(x) \) (c) \( \sin(x) \)

Q143. Use the double angle formula to find \( \sin(2x) \) if \( \sin(x) = \frac{3}{5} \) and \( \cos(x) = \frac{4}{5} \).

(a) \( \frac{12}{25} \) (b) \( \frac{24}{25} \) (c) \( \frac{6}{5} \)

Q144. Simplify using sum-to-product formulas: \( \sin(3x) + \sin(x) \).

(a) \( 2\sin(2x)\cos(x) \) (b) \( 2\cos(2x)\sin(x) \) (c) \( \sin(4x) \)

Q145. Prove the identity: \( \cos(2x) = \cos^2(x) - \sin^2(x) \). This is the:

(a) Sum formula (b) Difference formula (c) Double angle formula

Q146. In triangle ABC, if \( a = 8 \), \( b = 7 \), and \( \angle C = 60^\circ \), use the Law of Cosines to find side \( c \).

(a) \( \sqrt{113} \) (b) \( \sqrt{129} \) (c) \( \sqrt{65} \)

Q147. In triangle XYZ, \( \angle X = 30^\circ \), \( \angle Y = 45^\circ \), and \( x = 10 \). Use the Law of Sines to find side \( y \).

(a) \( 10\sqrt{2} \) (b) \( 10\sqrt{3} \) (c) \( \frac{10}{\sqrt{2}} \)

Q148. Given triangle PQR with \( p = 5 \), \( q = 7 \), \( r = 8 \), find \( \cos(R) \) using the Law of Cosines.

(a) \( \frac{1}{7} \) (b) \( \frac{11}{14} \) (c) \( -\frac{1}{7} \)

Q149. In triangle DEF, \( d = 15 \), \( e = 12 \), and \( \angle D = 60^\circ \). How many triangles can be formed (ambiguous case)?

(a) No triangle (b) One triangle (c) Two triangles

Q150. Given triangle ABC with \( \angle A = 22^\circ \), \( \angle B = 95^\circ \), and \( a = 15 \). Find side \( b \) using Law of Sines.

(a) \( \approx 25.4 \) (b) \( \approx 38.7 \) (c) \( \approx 10.3 \)

Q151. A surveyor needs to find the distance across a river. From point A on one bank, they sight a point B on the opposite bank. They then walk 50m downstream to point C and measure \( \angle ACB = 40^\circ \) and \( \angle CAB = 90^\circ \). Find the distance AB (width of the river).

(a) \( \approx 41.96 \) m (b) \( \approx 32.14 \) m (c) \( \approx 65.23 \) m

Q152. Two ships leave a port at the same time. Ship A sails at 20 mph in the direction N30°E, and Ship B sails at 25 mph in the direction S60°E. How far apart are the ships after 2 hours?

(a) \( \approx 40 \) miles (b) \( \approx 63 \) miles (c) \( \approx 52 \) miles

Q153. Find the area of triangle ABC if \( a = 10 \), \( b = 12 \), and \( \angle C = 30^\circ \).

(a) 30 sq units (b) 60 sq units (c) 120 sq units

Q154. Calculate the area of triangle PQR with sides \( p = 13 \), \( q = 14 \), and \( r = 15 \) using Heron's formula.

(a) 84 sq units (b) 42 sq units (c) 90 sq units

Q155. The area of triangle XYZ is \( 24 \) sq cm, and sides \( x = 6 \) cm and \( y = 8 \) cm. Find \( \sin(Z) \).

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

Q156. Solve \( 2\sin(x) + \sqrt{3} = 0 \) for \( x \) in \( [0, 2\pi) \).

(a) \( x = \frac{4\pi}{3}, \frac{5\pi}{3} \) (b) \( x = \frac{2\pi}{3}, \frac{4\pi}{3} \) (c) \( x = \frac{\pi}{3}, \frac{2\pi}{3} \)

Q157. Find the general solution for \( \tan(x) = \sqrt{3} \).

(a) \( x = \frac{\pi}{6} + n\pi \) (b) \( x = \frac{\pi}{3} + n\pi \) (c) \( x = \frac{\pi}{3} + 2n\pi \)

Q158. Solve for \( \cos(x) = 0 \) in the interval \( [0, 2\pi) \).

(a) \( x = \frac{\pi}{2}, \frac{3\pi}{2} \) (b) \( x = 0, \pi \) (c) \( x = \pi, 2\pi \)

Q159. Find all solutions for \( \tan^2(x) - 3 = 0 \) in \( [0, 2\pi) \).

(a) \( x = \frac{\pi}{3}, \frac{4\pi}{3} \) (b) \( x = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3} \) (c) \( x = \frac{2\pi}{3}, \frac{5\pi}{3} \)

Q160. Solve \( 2\cos(2x) - 1 = 0 \) for \( x \) in \( [0, \pi) \).

(a) \( x = \frac{\pi}{6}, \frac{5\pi}{6} \) (b) \( x = \frac{\pi}{6}, \frac{11\pi}{6} \) (c) \( x = \frac{\pi}{3}, \frac{5\pi}{3} \)

Q161. Find the solutions to \( \cos(3x) = -1 \) in \( [0, \pi) \).

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

Q162. Solve \( 2\sin^2(x) + 3\sin(x) + 1 = 0 \) for \( x \) in \( [0, 2\pi) \).

(a) \( x = \frac{7\pi}{6}, \frac{11\pi}{6} \) (b) \( x = \frac{\pi}{2}, \frac{3\pi}{2} \) (c) \( x = \frac{3\pi}{2}, \frac{7\pi}{6} \)

Q163. Find the solution(s) to \( \sin(2x) = \sin(x) \) in \( [0, 2\pi) \).

(a) \( x = 0, \frac{\pi}{3}, \pi, \frac{5\pi}{3} \) (b) \( x = 0, \frac{\pi}{3}, \pi, \frac{2\pi}{3} \) (c) \( x = \frac{\pi}{3}, \pi, \frac{5\pi}{3} \)

Q164. Solve \( \cos^2(x) = \cos(x) \) for \( x \) in \( [0, 2\pi) \).

(a) \( x = 0, \frac{\pi}{2}, \frac{3\pi}{2} \) (b) \( x = 0, \frac{\pi}{2}, \frac{3\pi}{2}, 2\pi \) (c) \( x = 0, \frac{\pi}{2}, \frac{3\pi}{2}, \frac{2\pi}{3} \)

Q165. Find the general solution of \( \cos(x) = -\frac{\sqrt{2}}{2} \).

(a) \( x = \pm \frac{3\pi}{4} + 2n\pi \) (b) \( x = \frac{3\pi}{4} + 2n\pi \) and \( x = \frac{5\pi}{4} + 2n\pi \) (c) \( x = \pm \frac{\pi}{4} + 2n\pi \)

Q166. Describe the transformation from \( y = \sin(x) \) to \( y = \sin(x - \frac{\pi}{4}) \).

(a) Shifted \( \frac{\pi}{4} \) units to the left (b) Shifted \( \frac{\pi}{4} \) units to the right (c) Stretched horizontally by a factor of \( \frac{\pi}{4} \)

Q167. Identify the amplitude of \( y = 0.5\cos(x) \).

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

Q168. Determine the period of \( y = \cos(\frac{1}{2}x) \).

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

Q169. Find the phase shift of \( y = 3\sin(2x + \pi) \).

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

Q170. Identify the vertical shift of \( y = -\cos(x) + 3 \).

(a) 3 units downward (b) 3 units upward (c) -3 units downward

Q171. What is the value of \( \arccos(1) \) in radians?

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

Q172. Evaluate \( \arcsin(-\frac{\sqrt{2}}{2}) \) in radians.

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

Q173. What is the domain of \( y = \arccos(x) \)?

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

Q174. Determine the range of \( y = \arctan(x) \).

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

Q175. Evaluate \( \arctan(\sqrt{3}) \) in radians.

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

Q176. Use the sum formula to find the exact value of \( \sin(105^\circ) \).

(a) \( \frac{\sqrt{6} - \sqrt{2}}{4} \) (b) \( \frac{\sqrt{6} + \sqrt{2}}{4} \) (c) \( \frac{\sqrt{2} - \sqrt{6}}{4} \)

Q177. Apply the difference formula to simplify \( \cos(x + \frac{\pi}{2}) \).

(a) \( \sin(x) \) (b) \( -\sin(x) \) (c) \( \cos(x) \)

Q178. Use the double angle formula to find \( \cos(2x) \) if \( \cos(x) = \frac{1}{3} \).

(a) \( \frac{2}{3} \) (b) \( -\frac{7}{9} \) (c) \( \frac{1}{9} \)

Q179. Simplify using product-to-sum formulas: \( 2\cos(3x)\sin(x) \).

(a) \( \sin(4x) - \sin(2x) \) (b) \( \sin(4x) + \sin(2x) \) (c) \( \cos(4x) - \cos(2x) \)

Q180. Prove the identity: \( \tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)} \). This is the:

(a) Sum formula (b) Difference formula (c) Double angle formula

Q181. In triangle ABC, if \( b = 10 \), \( c = 5 \), and \( \angle A = 45^\circ \), use the Law of Cosines to find side \( a \).

(a) \( \sqrt{125 - 50\sqrt{2}} \) (b) \( \sqrt{125 + 50\sqrt{2}} \) (c) \( \sqrt{75} \)

Q182. In triangle KLM, \( \angle K = 60^\circ \), \( \angle L = 60^\circ \), and \( k = 12 \). Use the Law of Sines to find side \( l \).

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

Q183. Given triangle RST with \( r = 10 \), \( s = 16 \), \( t = 14 \), find \( \cos(S) \) using the Law of Cosines.

(a) \( \frac{1}{2} \) (b) \( \frac{11}{14} \) (c) \( \frac{1}{4} \)

Q184. In triangle UVW, \( u = 20 \), \( v = 10 \), and \( \angle U = 30^\circ \). How many triangles can be formed (ambiguous case)?

(a) No triangle (b) One triangle (c) Two triangles

Q185. Given triangle DEF with \( \angle D = 35^\circ \), \( \angle E = 65^\circ \), and \( f = 20 \). Find side \( d \) using Law of Sines.

(a) \( \approx 14.3 \) (b) \( \approx 11.6 \) (c) \( \approx 24.5 \)

Q186. A plane flies on a bearing of 150° for 150 km and then turns and flies on a bearing of 60° for 100 km. Find the straight-line distance from the starting point to the final point.

(a) \( \approx 132 \) km (b) \( \approx 180 \) km (c) \( \approx 250 \) km

Q187. Two boats leave a dock at the same time. One boat travels due south at 15 mph. The other travels S70°E at 20 mph. How far apart are the boats after 3 hours?

(a) \( \approx 54 \) miles (b) \( \approx 63 \) miles (c) \( \approx 48 \) miles

Q188. Find the area of triangle DEF if \( d = 15 \), \( f = 20 \), and \( \angle E = 60^\circ \).

(a) \( 75\sqrt{3} \) sq units (b) \( 150 \) sq units (c) \( 150\sqrt{3} \) sq units

Q189. Calculate the area of triangle ABC with sides \( a = 24 \), \( b = 26 \), and \( c = 10 \) using Heron's formula.

(a) 120 sq units (b) 240 sq units (c) 60 sq units

Q190. The area of triangle RST is \( 48 \) sq cm, and sides \( r = 8 \) cm and \( t = 12 \) cm. Find \( \sin(S) \).

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

Q191. Solve \( 2\cos(x) + 1 = 0 \) for \( x \) in \( [0, 2\pi) \).

(a) \( x = \frac{2\pi}{3}, \frac{4\pi}{3} \) (b) \( x = \frac{\pi}{3}, \frac{5\pi}{3} \) (c) \( x = \frac{2\pi}{3}, \frac{5\pi}{3} \)

Q192. Find the general solution for \( \sin(x) = \frac{\sqrt{2}}{2} \).

(a) \( x = \frac{\pi}{4} + 2n\pi \) and \( x = \frac{3\pi}{4} + 2n\pi \) (b) \( x = \pm \frac{\pi}{4} + 2n\pi \) (c) \( x = \frac{\pi}{4} + n\pi \)

Q193. Solve for \( \tan(x) = 0 \) in the interval \( [0, 2\pi) \).

(a) \( x = 0, \pi \) (b) \( x = \pi, 2\pi \) (c) \( x = 0, 2\pi \)

Q194. Find all solutions for \( \cos^2(x) - \cos(x) = 0 \) in \( [0, 2\pi) \).

(a) \( x = \frac{\pi}{2}, \frac{3\pi}{2} \) (b) \( x = 0, \frac{\pi}{2}, \frac{3\pi}{2} \) (c) \( x = 0, \frac{\pi}{2}, \frac{3\pi}{2}, 2\pi \)

Q195. Solve \( \sin(2x) = 1 \) for \( x \) in \( [0, \pi) \).

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

Q196. Find the solutions to \( \tan(2x) = 0 \) in \( [0, \pi) \).

(a) \( x = 0, \frac{\pi}{2} \) (b) \( x = 0, \frac{\pi}{4} \) (c) \( x = \frac{\pi}{2}, \pi \)

Q197. Solve \( 2\cos^2(x) + \cos(x) - 1 = 0 \) for \( x \) in \( [0, 2\pi) \).

(a) \( x = 0, \frac{2\pi}{3}, \frac{4\pi}{3} \) (b) \( x = 0, \frac{\pi}{3}, \frac{5\pi}{3} \) (c) \( x = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3} \)

Q198. Find the solution(s) to \( \tan(x) = -\sqrt{3} \) in \( [0, 2\pi) \).

(a) \( x = \frac{2\pi}{3}, \frac{5\pi}{3} \) (b) \( x = \frac{2\pi}{3}, \frac{4\pi}{3} \) (c) \( x = \frac{4\pi}{3}, \frac{5\pi}{3} \)

Q199. Solve \( \tan^2(x) = 3 \) for \( x \) in \( [0, 2\pi) \).

(a) \( x = \frac{\pi}{3}, \frac{4\pi}{3} \) (b) \( x = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3} \) (c) \( x = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{5\pi}{3}, \frac{7\pi}{3} \)

Q200. Find the general solution of \( \tan(x) = 1 \).

(a) \( x = \frac{\pi}{4} + 2n\pi \) (b) \( x = \frac{\pi}{4} + n\pi \) (c) \( x = \pm \frac{\pi}{4} + n\pi \)