9709 · 2.3
Trigonometry flashcards
Revision flashcards for Cambridge 9709 Trigonometry (syllabus 2.3). Flip, recall, then mark a real past-paper question.
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Define $\sec\theta$, $\csc\theta$, and $\cot\theta$.
$\sec\theta = \frac{1}{\cos\theta}$, $\csc\theta = \frac{1}{\sin\theta}$, $\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$.
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State the Pythagorean identity involving $\tan\theta$ and $\sec\theta$.
$1 + \tan^2\theta \equiv \sec^2\theta$. This is derived by dividing $\sin^2\theta + \cos^2\theta \equiv 1$ by $\cos^2\theta$.
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State the Pythagorean identity involving $\cot\theta$ and $\csc\theta$.
$1 + \cot^2\theta \equiv \csc^2\theta$. This is derived by dividing $\sin^2\theta + \cos^2\theta \equiv 1$ by $\sin^2\theta$.
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What is the double angle formula for $\sin(2A)$?
$\sin(2A) = 2\sin A \cos A$. It is derived from $\sin(A+B)$ by setting $B=A$.
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List the three double angle formulae for $\cos(2A)$.
1. $\cos(2A) = \cos^2 A - \sin^2 A$ 2. $\cos(2A) = 2\cos^2 A - 1$ 3. $\cos(2A) = 1 - 2\sin^2 A$
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When expressing $a\sin\theta + b\cos\theta$ as $R\sin(\theta+\alpha)$, what is the formula for $R$?
$R = \sqrt{a^2 + b^2}$. $R$ is always positive and represents the amplitude of the combined wave.
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When expressing $a\sin\theta + b\cos\theta$ as $R\sin(\theta+\alpha)$, how do you find $\alpha$?
By expanding $R\sin(\theta+\alpha) = R(\sin\theta\cos\alpha + \cos\theta\sin\alpha)$ and comparing coefficients, we get $a=R\cos\alpha$ and $b=R\sin\alpha$. Therefore, $\tan\alpha = \frac{b}{a}$.
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Common Trap: After finding the principal value for a trig equation, what must you do next?
You must find all other solutions within the specified interval. Use the CAST diagram or graph symmetry. For example, if $\sin x = k$, other solutions are $180^\circ - x$ (or $\pi - x$ in radians), plus multiples of $360^\circ$ (or $2\pi$).
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Common Trap: When solving an equation like $\cos(2x) = 0.5$ for $0^\circ \le x \le 360^\circ$, what is the first step?
Adjust the interval for the 'new' angle. The interval for $2x$ is $0^\circ \le 2x \le 720^\circ$. Find all solutions for $2x$ in this larger interval before dividing by 2 to find $x$.
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Common Trap: What can happen if you square both sides of a trigonometric equation?
Squaring can introduce extraneous (false) solutions. For example, if $\sin x = 0.5$, squaring gives $\sin^2 x = 0.25$. This equation is also satisfied by $\sin x = -0.5$, which was not a solution to the original. Always check your final answers in the original equation if you square it.
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How do you decide which version of $\cos(2A)$ to use?
Look at the other terms in the equation. To create a quadratic in $\cos A$, use $\cos(2A) = 2\cos^2 A - 1$. To create a quadratic in $\sin A$, use $\cos(2A) = 1 - 2\sin^2 A$.