Cards In This Deck

• $\text{First step to sketch the graph of }y=cosec2\theta \text{?}$
  $\text{Sketch the graph of }y=\sin 2\theta \text{.}$
• $\text{How do you evaluate }cosec(210^{\circ })\text{ without a calculator?}$
  $\text{Find }\sin (210^{\circ })\text{ first, then take the reciprocal.}$
• $$\begin{gathered} \text{True or False?} \\ \text{The solution to }\cot \theta =0\text{ is the same as the solution to }\tan \theta =0\text{.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: It implies }\tan \theta \text{ is undefined (or }\cos \theta =0\text{).} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{The equation }\sec x=0.5\text{ has two solutions in the interval }0\le x\le 360^{\circ }\text{.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: It has no solutions.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \cot x\text{ is undefined when }\cos x=0\text{.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: }\cot x\text{ is undefined when }\sin x=0\text{.} \end{gathered}$$
• $\text{What values of }x\text{ must be excluded from the domain of }y=\sec x\text{?}$
  $$\begin{gathered} x=\frac{(2n+1) \\ \pi }{2}\text{ (Odd multiples of }\frac{ \\ \pi }{2}\text{ or }90^{\circ }\text{)} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{The function }y=\arcsin x\text{ is defined for all }x\in \mathbb{R}\text{.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: The domain is }-1\le x\le 1\text{.} \end{gathered}$$
• $\text{What is the principal value range (output) for }y=\arccos x\text{ in radians?}$
  $$\begin{gathered} 0\le y\le \\ \pi \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \arcsin x=(\sin x{)}^{-1}=\frac{1}{\sin x} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: }\frac{1}{\sin x}=cosecx \end{gathered}$$
• $\text{Which identity should be substituted to solve the quadratic equation }3{\tan }^2\theta +5\sec \theta +1=0\text{?}$
  ${\tan }^2\theta ={\sec }^2\theta -1$
• $$\begin{gathered} \text{True or False?} \\ {\sec }^{2}x-{\tan }^{2}x=-1 \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: }{\sec }^{2}x-{\tan }^{2}x=1 \end{gathered}$$
• $\text{To prove the identity }1+{\cot }^{2}x\equiv {cosec}^{2}x\text{, what operation do you perform on the standard identity }{\sin }^{2}x+{\cos }^{2}x\equiv 1\text{?}$
  $\text{Divide every term by }{\sin }^{2}x\text{.}$
• $\text{To prove the identity }\tan 2\theta =\frac{2}{\cot \theta -\tan \theta }\text{, what is the most efficient first step?}$
  $\text{Use the double angle formula for }\tan 2\theta \text{: }\frac{2\tan \theta }{1-{\tan }^2\theta }\text{ and divide numerator and denominator by }\tan \theta \text{.}$
• $$\begin{gathered} \text{True or False?} \\ \cos 2A\equiv 1+2{\sin }^{2}A \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Identity: }\cos 2A\equiv 1-2{\sin }^{2}A \end{gathered}$$
• $\text{What is the standard procedure to solve an equation involving }\sin 2\theta \text{ and }\cos 2\theta \text{ terms (e.g., }\sin 2\theta +\cos 2\theta =0\text{)?}$
  $\text{Divide by }\cos 2\theta \text{ to form an equation in }\tan 2\theta$
• $$\begin{gathered} \text{True or False?} \\ \text{To solve }\sin 2\theta =\sin \theta \text{, you should divide both sides by }\sin \theta \text{ to get }2\cos \theta =1\text{.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Method: Rearrange to }\sin 2\theta -\sin \theta =0\text{, then factorise: }\sin \theta (2\cos \theta -1)=0\text{.} \end{gathered}$$
• $\text{When solving the equation }3\cos 2x-\cos x+2=0\text{, which expansion of }\cos 2x\text{ should you choose?}$
  $2{\cos }^{2}x-1$
• $\text{If you need to find the exact value of }\sin 15^{\circ }\text{ without a calculator, how should you rewrite }15^{\circ }\text{?}$
  $\text{Rewrite as }(45^{\circ }-30^{\circ })\text{ or }(60^{\circ }-45^{\circ })$
• $$\begin{gathered} \text{True or False?} \\ \sin (A+B)\equiv \sin A+\sin B \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Identity: }\sin (A+B)\equiv \sin A\cos B+\cos A\sin B \end{gathered}$$
• $\text{Formula for }\tan (A-B)$
  $\frac{\tan A-\tan B}{1+\tan A\tan B}$
• $$\begin{gathered} \text{True or False?} \\ \text{If }f(x)=12\cos x+5\sin x\text{, the maximum value is }12+5=17\text{.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: }R=\sqrt{12^2+5^2}=13 \end{gathered}$$
• $\text{What is the first step to express }3\sin x+4\cos x\text{ in the form }R\sin (x+\alpha )\text{?}$
  $$\begin{gathered} \text{Expand }R\sin (x+\alpha )\text{ using the addition formula:} \\ R\sin x\cos \alpha +R\cos x\sin \alpha \end{gathered}$$
• $\text{When solving }a\cos \theta +b\sin \theta =c\text{ using the harmonic form }R\cos (\theta -\alpha )\text{, what is the condition for real solutions to exist?}$
  $|c|⩽R\text{ (or }{c}^2⩽a^2+b^2\text{)}$