Level 4: Advanced Trigonometry

Q201. Solve \(2\cos(3x) + \sqrt{3} = 0\) on \([0, 2\pi)\).

(a) \( x = \frac{\pi}{9}, \frac{7\pi}{9} \) (b) \( x = \frac{5\pi}{9}, \frac{11\pi}{9} \) (c) \( x = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6} \)

Q202. Solve \( \tan(2x) = -\sqrt{3} \) in the interval \([0, 2\pi)\).

(a) \( x = \frac{5\pi}{12}, \frac{11\pi}{12}, \frac{17\pi}{12}, \frac{23\pi}{12} \) (b) \( x = \frac{\pi}{3}, \pi \) (c) \( x = \frac{5\pi}{6}, \frac{7\pi}{6} \)

Q203. Simplify using product-to-sum: \( \sin(x)\cos(2x) \)

(a) \( \frac{1}{2}[\sin(3x) + \sin(x)] \) (b) \( \frac{1}{2}[\sin(3x) - \sin(x)] \) (c) \( \sin(3x) \)

Q204. Solve \( \sin(2x) + \sin(x) = 0 \) on \([0, 2\pi)\).

(a) \( x = 0, \pi, \frac{2\pi}{3}, \frac{4\pi}{3} \) (b) \( x = 0, \pi, \frac{\pi}{2}, \frac{3\pi}{2} \) (c) \( x = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6} \)

Q205. Evaluate \( \arcsin(\sin(\frac{7\pi}{4})) \) in the principal range.

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

Q206. Use the identity to simplify: \( \sin(4x) - \sin(2x) \)

(a) \( 2\cos(3x)\sin(x) \) (b) \( 2\cos(2x)\sin(x) \) (c) \( 2\cos(3x)\sin(3x) \)

Q207. Solve \( \cos(3x) = -\frac{1}{2} \) over \([0, 2\pi)\).

(a) \( x = \frac{5\pi}{9}, \frac{7\pi}{9}, \frac{14\pi}{9}, \frac{16\pi}{9} \) (b) \( x = \frac{\pi}{9}, \frac{11\pi}{9}, \frac{19\pi}{9} \) (c) \( x = \frac{2\pi}{9}, \frac{8\pi}{9}, \frac{10\pi}{9}, \frac{14\pi}{9} \)

Q208. Find the exact value of \( \cot\bigl(\frac{13\pi}{12}\bigr) \) using sum formulas.

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

Q209. Solve \( 2\sin^2(x) - \sqrt{3}\sin(x) + \tfrac{1}{2} = 0 \) in \([0, 2\pi)\).

(a) \( x = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6} \) (b) \( x = \frac{\pi}{2}, \frac{3\pi}{2} \) (c) \( x = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3} \)

Q210. Determine the period of \( y = \sin\bigl(\frac{3}{2}x\bigr) \) and identify any phase shift.

(a) Period=\(\frac{4\pi}{3}\), no shift (b) Period=\(\frac{2\pi}{3}\), shift=\(\frac{\pi}{2}\) (c) Period=\(\frac{2\pi}{3}\), no shift

Q211. Solve \( 4\sin^2(x) - 3= 0 \) for \( x\in [0,2\pi).\)

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

Q212. Use sum-to-product to simplify \( \cos(5x)+ \cos(3x) \).

(a) \( 2\cos(4x)\cos(x) \) (b) \( 2\sin(4x)\sin(x) \) (c) \( \cos(8x) \)

Q213. Solve \( \sin(3x)= \sin(x) \) on \([0, 2\pi).\)

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

Q214. Find the exact value of \( \sin\bigl(\frac{5\pi}{12}\bigr) \) using half-angle or sum formulas.

(a) \( \frac{\sqrt{6} + \sqrt{2}}{4} \) (b) \( \frac{\sqrt{6} - \sqrt{2}}{4} \) (c) \( \frac{\sqrt{3}}{2} \)

Q215. Solve \( \tan(4x) = 1 \) in \([0, 2\pi).\)

(a) \( x= \frac{\pi}{16}, \frac{3\pi}{16}, ...\) (b) \( x= \frac{\pi}{8}, \frac{3\pi}{8}, \frac{5\pi}{8}, \frac{7\pi}{8}\) (c) \( x= \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\)

Q216. Use the identity \( \cos(2x)= 1 - 2\sin^2(x) \) to rewrite \( 3\cos(2x)-1 \).

(a) \( 3(1-2\sin^2 x)-1= 2-6\sin^2 x\) (b) \( 3(1-2\sin^2 x)-1= 2 - 3\sin^2 x\) (c) \( 1- 6\cos^2 x\)

Q217. Solve \( 2\cos^2(x)- 2\cos(x)-1=0 \) on \([0,2\pi).\)

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

Q218. Use sum formulas: \( \sin\bigl(\frac{5\pi}{4}+x\bigr)\).

(a) \( -\frac{\sqrt{2}}{2}\cos x - \frac{\sqrt{2}}{2}\sin x \) (b) \( -\sin(x+\frac{\pi}{4}) \) (c) \( \sin(x+\frac{\pi}{2})\)

Q219. Solve \( \sin(2x)= \sqrt{3}\cos(x) \) in \([0,2\pi).\)

(a) \( x= \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}\) (b) \( x= \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}\) (c) \( x= \frac{\pi}{2}, \frac{3\pi}{2}\)

Q220. Evaluate: \( \arctan(\tan(\frac{13\pi}{6})) \).

(a) \( \frac{\pi}{6}\) (b) \( -\frac{\pi}{6}\) (c) \( \frac{13\pi}{6}\)

Q221. Use difference formula: \( \cos\bigl(x - \frac{2\pi}{3}\bigr) \).

(a) \( \cos x\cos\frac{2\pi}{3} + \sin x\sin\frac{2\pi}{3}\) (b) \( \cos x\cos\frac{2\pi}{3} - \sin x\sin\frac{2\pi}{3}\) (c) \( \cos x+ \sin x\)

Q222. Solve \( \cos(2x)= \sin(3x) \) on \([0,2\pi).\)

(a) no solutions (b) \( x= \frac{\pi}{6}, \frac{\pi}{2}, ...\) (c) infinitely many solutions in [0,2\pi)

Q223. Prove the identity: \( 1+ \tan^2 A= \sec^2 A\).

(a) Double-angle formula (b) Pythagorean identity (c) Sum to product

Q224. Solve \( \sin(x)+ \cos(x)=1 \) for \( x\in[0,2\pi).\)

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

Q225. Find the amplitude & period of \( y= -2\cos(3x+ \pi) \).

(a) Amplitude 2, period \( \frac{2\pi}{3}\) (b) Amplitude 1, period \( \frac{2\pi}{3}\) (c) Amplitude 2, period \(2\pi\)

Q226. Evaluate \( \arccos\bigl(-\frac{\sqrt{3}}{2}\bigr) \).

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

Q227. Solve \( \sin(4x) = \frac{1}{2} \) on \([0,2\pi).\)

(a) \( x= \frac{\pi}{24}, \frac{5\pi}{24}, \frac{13\pi}{24}, \frac{17\pi}{24}, ...\) (b) \( x= \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}\) (c) no solutions

Q228. Use half-angle: \( \sin^2\bigl(\frac{x}{2}\bigr)= \frac{1-\cos x}{2}\).

(a) correct identity (b) no, it should be \(\frac{1+\cos x}{2}\) (c) not an identity

Q229. Compute \( \tan\bigl(\frac{17\pi}{12}\bigr)\) using sum formula ( \( \frac{17\pi}{12}= \frac{3\pi}{4}+ \frac{\pi}{6}\)).

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

Q230. Solve \( \tan^2(x)=3\) on \([0,2\pi).\)

(a) \( x= \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}\)\" (b) \( x= \frac{\pi}{6}, \frac{7\pi}{6}\)\" (c) \( x= \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\)\"

Q231. Evaluate: \( \arcsin(-1)\).

(a) \( -\frac{\pi}{2}\) (b) \( \frac{3\pi}{2}\) (c) 0

Q232. Find domain of \( y= \arccot(x)\).

(a) x \( \in (-\infty, \infty)\) (b) x \( \in [0,\infty)\) (c) none

Q233. Solve \( 3\cos(2x)-2=0 \) in \([0,2\pi).\)

(a) no solutions (b) \( x= \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}\) (c) \( x= \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}\)

Q234. Use sum formula: \( \cos\bigl(\frac{7\pi}{12}\bigr)\).

(a) \( \frac{\sqrt{6}+ \sqrt{2}}{4}\) (b) \( \frac{\sqrt{6}- \sqrt{2}}{4}\) (c) \( \frac{\sqrt{3}}{2}\)

Q235. Solve \( \sin(2x)= -\frac{\sqrt{2}}{2}\) in \([0,2\pi).\)

(a) \( x= \frac{3\pi}{8}, \frac{7\pi}{8}, \frac{11\pi}{8}, \frac{15\pi}{8}\) (b) \( x= \frac{5\pi}{4}, \frac{7\pi}{4}\) (c) \( x= \frac{\pi}{6}, \frac{5\pi}{6}\)

Q236. Find \( \tan(3x) \) if \( \tan(x)=\frac{2}{3}\).

(a) \( \frac{3\tan x - \tan^3 x}{1-3\tan^2 x} \) (b) 1 (c) none

Q237. Solve \( 2\sin^2(x)- 3\sin(x)+1=0 \) in \([0,2\pi).\)

(a) x=0, \(\frac{\pi}{6}, \frac{5\pi}{6}\) (b) x= \( \frac{\pi}{2}, \frac{3\pi}{2}\) (c) x= \( \frac{\pi}{3}, \frac{2\pi}{3}, \pi\)

Q238. Use the formula \( \sin(A)-\sin(B)= 2\cos\bigl(\frac{A+B}{2}\bigr)\sin\bigl(\frac{A-B}{2}\bigr)\).

(a) correct (b) incorrect (c) only for A=B

Q239. Solve \( \cos(3x)= \sqrt{\frac{1}{2}}\) in \([0,2\pi).\)

(a) no solutions (b) x= \( \frac{\pi}{12}, \frac{5\pi}{12}, ...\) (c) infinite solutions in [0,2\pi)

Q240. Rewrite \( \sin^2(x)= \frac{1-\cos(2x)}{2}\).

(a) standard identity (b) not correct (c) only for x>0

Q241. Compute \( \arcsin\bigl(\frac{\sqrt{3}}{2}\bigr)\).

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

Q242. Solve \( \tan(5x)=0\) in \([0,2\pi).\)

(a) x= multiple of \( \frac{\pi}{5}\)\" (b) x= 0, \( \frac{\pi}{5}, \frac{2\pi}{5}, ...\) (c) no solutions

Q243. Use difference formula: \( \sin(2x- \frac{\pi}{4})\).

(a) \( \sin(2x)\cos(\frac{\pi}{4})- \cos(2x)\sin(\frac{\pi}{4})\) (b) \( 2\sin(x-\frac{\pi}{8})\cos(x-\frac{\pi}{8})\) (c) none

Q244. Solve \( \sin(x)= \cos(x)\) in \([0,2\pi).\)

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

Q245. Solve \( \sin(x)= -\cos(x)\) in \([0,2\pi).\)

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

Q246. Solve \( \tan(2x)= \cot(x)\) on \([0,2\pi).\)

(a) x= \( \frac{\pi}{6}, \frac{\pi}{2}, ...\) (b) x= \( \frac{\pi}{4}, \pi \) (c) x= no solutions

Q247. Find the amplitude of \( y= -3\sin(4x)\).

(a) 3 (b) 4 (c) 1

Q248. Find the phase shift of \( y= \sin(2x- \frac{\pi}{3})\).

(a) shift= \( \frac{\pi}{6} \) right (b) shift= \( \frac{\pi}{6} \) left (c) no shift

Q249. Rewrite using sum-to-product: \( \cos(3x)+ \cos(x)\).

(a) \( 2\cos(2x)\cos(x)\) (b) \( 2\sin(2x)\sin(x)\) (c) none

Q250. Solve \( \cos(2x)=0\) in \([0,2\pi).\)

(a) x= \( \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\) (b) x= \( \frac{\pi}{2}, \frac{3\pi}{2}\) (c) none

Q251. Evaluate \( \arccos(-1)\).

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

Q252. Solve \( \csc(x)= 2\) in \([0,2\pi).\)

(a) x= \( \frac{\pi}{6}, \frac{5\pi}{6}\) (b) x= \( \frac{\pi}{2}, \frac{3\pi}{2}\) (c) x= no solutions

Q253. Use the identity: \( \sin(3x)=3\sin(x)-4\sin^3(x)\).

(a) correct triple-angle (b) not correct (c) belongs to sum-of-angles

Q254. Solve \( \tan(x+ \frac{\pi}{4})=1\) in \([0,2\pi).\)

(a) x= 0, \( \pi\) (b) x= \( \frac{\pi}{4}, \frac{5\pi}{4}\) (c) x= \( \frac{\pi}{6}, \frac{7\pi}{6}\)

Q255. Compute \( \sec^2(x)- \tan^2(x)\).

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

Q256. Solve \( 3\cos(4x)-1=0\) in \([0,2\pi).\)

(a) x= multiple of \( \frac{\pi}{8}...\) (b) no solutions (c) x= \( \frac{\pi}{4}, \frac{3\pi}{4}, ...\)

Q257. Evaluate \( \arctan(\sqrt{3})\).

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

Q258. Solve \( \sin(3x)= -\frac{\sqrt{2}}{2}\) in \([0,2\pi).\)

(a) x= \( \frac{7\pi}{12}, \frac{11\pi}{12}, ...\) (b) x= \( \frac{5\pi}{4}, \frac{7\pi}{4}\) (c) none

Q259. Factor: \( \sin(x)\cos(x)- \sin(x)=0\).

(a) \( \sin(x)[\cos(x)-1]=0\) (b) no factor (c) none

Q260. Solve \( \cos^2(x)= \frac{1}{4}\) on \([0,2\pi).\)

(a) x= \( \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}\) (b) x= \( \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}\) (c) none

Q261. Solve \( \cos(2x)- \sin(x)=0\) in \([0,2\pi).\)

(a) x= ... multiple solutions (b) x= \( \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\) (c) none

Q262. Rewrite \( 1-\cos^2(x)= \sin^2(x)\).

(a) standard pyth identity (b) no (c) only for x>0

Q263. Compute \( \tan\bigl(\frac{5\pi}{12}\bigr)\).

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

Q264. Solve \( \tan(x)= -\sqrt{3}\) in \([0,2\pi).\)

(a) x= \( \frac{2\pi}{3}, \frac{5\pi}{3}\) (b) x= \( \frac{\pi}{3}, \frac{4\pi}{3}\) (c) x= \( \frac{\pi}{6}, \frac{7\pi}{6}\)

Q265. Evaluate \( \arccos(\frac{1}{2})\).

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

Q266. Solve \( 2\sin(2x)- 1=0\) in \([0,2\pi).\)

(a) x= \( \frac{\pi}{12}, \frac{5\pi}{12}, \frac{7\pi}{12}, \frac{11\pi}{12}...\) (b) x= \( \frac{\pi}{8}, \frac{5\pi}{8}, ...\) (c) x= \( \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}\)

Q267. Compute \( \csc\bigl(\frac{5\pi}{6}\bigr)\).

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

Q268. Solve \( \sin(4x)= \sin(2x)\) on \([0,2\pi).\)

(a) x=0, \( \frac{\pi}{6}, \frac{\pi}{2}, ... (b) x= \( \frac{\pi}{4}, \frac{3\pi}{4}, ... (c) no solutions

Q269. Solve \( \cos(3x)= \frac{1}{2}\) in \([0,2\pi).\)

(a) x= \( \frac{\pi}{9}, \frac{5\pi}{9}, \frac{13\pi}{9}, \frac{17\pi}{9}\) (b) none (c) x= \( \frac{\pi}{6}, \frac{5\pi}{6}\)

Q270. Solve \( \tan^2(x)= 1\) on \([0,2\pi).\)

(a) x= \( \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\) (b) x= \( \frac{\pi}{6}, \frac{5\pi}{6}\) (c) none

Q271. Use double-angle to rewrite \( \sin(2x)\) in terms of \( \tan(x)\).

(a) \( \frac{2\tan(x)}{1+ \tan^2(x)}\) (b) none (c) \(2\sin x\cos x\)

Q272. Compute \( \sin\bigl(\frac{11\pi}{12}\bigr)\).

(a) \( \frac{\sqrt{6}+ \sqrt{2}}{4}\) (b) \( \frac{\sqrt{6}- \sqrt{2}}{4}\) (c) 1

Q273. Solve \( \sin(x)- \sin(2x)=0\) on \([0,2\pi).\)

(a) x=0, \( \frac{\pi}{2}, \pi, \frac{3\pi}{2}\) (b) x= 0, \( \pi, \frac{2\pi}{3}, \frac{4\pi}{3}\) (c) none

Q274. Solve \( \tan(2x)= 2\tan(x)\) in \([0,2\pi).\)

(a) all x (b) x= 0, \( \frac{\pi}{2}, \pi, \frac{3\pi}{2}\) (c) no solutions

Q275. Rewrite \( 1- 2\sin^2(x)= \cos(2x)\).

(a) standard identity (b) none (c) only for x>0

Q276. Evaluate \( \arccos(0)\).

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

Q277. Solve \( \sin(4x)=1\) on \([0,2\pi).\)

(a) x= \( \frac{\pi}{8}, \frac{3\pi}{8}, ... (b) x= \( \frac{\pi}{4}\) (c) no solutions

Q278. Find domain of \( y= \arctan(x)\).

(a) all real x (b) none (c) x>0 only

Q279. Rewrite \( \cos(3x)=4\cos^3(x)-3\cos(x)\).

(a) standard triple angle (b) incorrect (c) partial sum

Q280. Solve \( 2\cos(2x)+ 1=0\) in \([0,2\pi).\)

(a) x= \( \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}\) (b) x= \( \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}\) (c) x= none

Q281. Compute \( \arcsin(1)\).

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

Q282. Solve \( \cos(x)= -\sin(x)\) in \([0,2\pi).\)

(a) x= \( \frac{3\pi}{4}, \frac{7\pi}{4}\) (b) x= \( \frac{5\pi}{4}, \frac{3\pi}{4}\) (c) x=0, \( \pi\)

Q283. Simplify \( \tan^2(x)- \sec^2(x)\).

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

Q284. Solve \( \sin(2x)= \sin(x)\) in \([0,2\pi).\)

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

Q285. Use difference formula: \( \cos(2x- \frac{\pi}{6})\).

(a) \( \cos(2x)\cos(\frac{\pi}{6})+ \sin(2x)\sin(\frac{\pi}{6})\) (b) none (c) \( \cos(2x-\frac{\pi}{3})\)

Q286. Solve \( 3\sin(x)- \sqrt{3}=0\) in \([0,2\pi).\)

(a) x= \( \frac{\pi}{3}, \frac{2\pi}{3}\) (b) x= \( \frac{\pi}{6}, \frac{5\pi}{6}\) (c) x= none

Q287. Compute \( \cot\bigl(\frac{5\pi}{12}\bigr)\).

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

Q288. Find domain of \( y= \arcsin(x)\).

(a) x\(\in[-1,1]\) (b) x\(\in[0,\infty)\) (c) none

Q289. Solve \( \tan(3x)=0\) 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= none (c) x= \( \frac{\pi}{4}, \frac{3\pi}{4}...\)

Q290. Solve \( \cos(4x) + \sin(2x) = 0 \) on \([0, 2\pi)\).

(a) \( x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \) (b) \( x = \frac{\pi}{8}, \frac{3\pi}{8}, \frac{5\pi}{8}, \frac{7\pi}{8}, ... \) (c) \( x = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6} \)