Flashcards · 9 cards

Trigonometry

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Rule
How many radians make $180^\circ$ ? The arc length of a sector?
Answer
$180^\circ = \pi$ radians. Arc length $s = r\theta$ , with $\theta$ in radians.
A sectorarc length s = rθ (θ in radians); 180° = π rad
From the lesson: Radians, compound and double angles
Rule
$\sin(A \pm B)$ and $\cos(A \pm B)$ ?
Answer
$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$ . $\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$ : the sign flips.
The angle A + Bsin(A ± B) = sin A cos B ± cos A sin B
From the lesson: Radians, compound and double angles
Know it
$\tan(A + B)$ ?
Answer
$\dfrac{\tan A + \tan B}{1 - \tan A\tan B}$ .
From the lesson: Radians, compound and double angles
Know it
$\sin 2A$ and the three forms of $\cos 2A$ ?
Answer
$\sin 2A = 2\sin A\cos A$ . $\cos 2A = \cos^2 A - \sin^2 A = 1 - 2\sin^2 A = 2\cos^2 A - 1$ .
From the lesson: Radians, compound and double angles
Know it
An equation has both $\sin^2 x$ and $\cos x$ in it. How do you get it into one ratio?
Answer
Use $\sin^2 x = 1 - \cos^2 x$ (from $\sin^2 x + \cos^2 x = 1$ ). It becomes a quadratic in $\cos x$ : solve that, then find $x$ .
From the lesson: Trigonometric equations
Rule
Solve $\sin x = -0.5$ for $0^\circ \le x \le 360^\circ$ .
Answer
The reference angle is $30^\circ$ . Sine is negative in the third and fourth quadrants: $x = 180^\circ + 30^\circ = 210^\circ$ and $x = 360^\circ - 30^\circ = 330^\circ$ .
Where each ratio is positiveAll, Sin, Tan, Cos: going anticlockwise from 0°
From the lesson: Trigonometric equations
Know it
Solving a trig equation in a range: how do you make sure you have every answer?
Answer
For each value of the ratio, find the reference angle, then every quadrant in the range where the ratio has that sign.
From the lesson: Trigonometric equations
Which method?
WAEC 2018 · Paper 2 · Q1
If $\dfrac{\tan\theta + \tan30^\circ}{1 - \tan\theta\tan30^\circ} + 1 = 0$ , find $\tan\theta$ , leaving the answer in surd form.
What does the fraction remind you of?
Answer
The formula for $\tan(\theta + 30^\circ)$ . So the equation says $\tan(\theta + 30^\circ) = -1$ . (Or put $\tan 30^\circ = \frac{1}{\sqrt3}$ and solve for $\tan\theta$ .)
From the lesson: Radians, compound and double angles Try the question
Which method?
WAEC 2022 · Paper 2 · Q4
Solve $3\cos2x - \sin x = 0$ for $0^\circ \le x \le 360^\circ$ (2 d.p.).
Which form of $\cos 2x$ do you use?
Answer
$1 - 2\sin^2 x$ , so that everything is in $\sin x$ : the equation becomes a quadratic in $\sin x$ .
From the lesson: Trigonometric equations Try the question