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NSWMaths AdvancedQuick questions
Year 12: Trigonometric Functions
Quick questions on Solving trigonometric equations: principal values, multiple angles and quadratics in sine and cosine
15short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.
What is principal value and all solutions?Show answer
The principal value is the standard "calculator" inverse:
What is quadrants and signs?Show answer
The signs of the trig functions in each quadrant ("All Stations To Central" or ASTC):
What is multiple-angle equations?Show answer
For $\sin(b x) = k$ on $0 \le x \le L$, substitute $u = b x$. The interval for $u$ becomes $0 \le u \le b L$, which contains $b$ times as many solutions. Find all $u$ in that expanded interval, then divide each by $b$ to get the $x$ values.
What is reducing with identities?Show answer
If an equation mixes trig functions, use identities to reduce to one. Common moves:
What is quadratics in $\sin$ or $\cos$?Show answer
An equation like $2 \sin^2 x - 3 \sin x + 1 = 0$ is a quadratic in $u = \sin x$. Factor or use the quadratic formula:
What is basic sine equation?Show answer
Solve $\sin x = -\frac{\sqrt{3}}{2}$ for $0 \le x \le 2 \pi$.
What is cosine equation?Show answer
Solve $\cos x = -\frac{1}{2}$ for $0 \le x \le 2 \pi$.
What is multiple angle?Show answer
Solve $\sin 3x = 0$ for $0 \le x \le \pi$.
What is reducing with Pythagoras?Show answer
Solve $2 \cos^2 x + \sin x - 1 = 0$ for $0 \le x \le 2 \pi$.
What is using a double angle to factor?Show answer
Solve $\sin 2 x = \cos x$ for $0 \le x \le 2 \pi$.
What is reporting only the principal value?Show answer
A question with a given interval expects all solutions in that interval, not just one. Always sweep through quadrants and through periodic shifts.
What is forgetting to expand the interval for multiple angles?Show answer
If $u = 2 x$ and $x \in [0, 2\pi]$, then $u \in [0, 4\pi]$. Missing this halves the number of solutions found.
What is dividing by $\cos x$ without checking?Show answer
Dividing by $\cos x$ to introduce $\tan x$ loses the solutions where $\cos x = 0$. Either check those separately or factor instead of dividing.
What is accepting $|\sin x| > 1$ or $|\cos x| > 1$?Show answer
A quadratic in $\sin$ may produce a root outside $[-1, 1]$. Reject those, do not try to solve.
What is wrong quadrants from the wrong identity?Show answer
If $\sin x < 0$ and you "find" $x$ in Q1 or Q2, you have the wrong sign. Always use the quadrant rule (ASTC) to place the solution.