Extended. The calculator gives you one angle. The graph tells you where the others are. Solving sin x = 2/3 for 0° ≤ x ≤ 360° means finding every angle, not just the first.
Rearrange to \(\sin x = k\), \(\cos x = k\) or \(\tan x = k\). Use the inverse key for the first angle, then the graph's symmetry for the rest: \(\sin\): \(180^\circ - \alpha\); \(\cos\): \(360^\circ - \alpha\); \(\tan\): \(\alpha + 180^\circ\). Keep only answers in \(0^\circ \le x \le 360^\circ\).
Fig. 1: the line y = 2/3 crosses y = sin x twice between 0° and 360°, at x = 41.8° and x = 180° − 41.8° = 138.2° (plotted from computed values)
Section 01
Every solution in the range.
In the lab, drag the horizontal line \(y = k\) up and down across the graph and watch the solutions move in pairs. Notice when there are two solutions, one, or none at all.
Step 1, rearrange: get \(\sin x\), \(\cos x\) or \(\tan x\) on its own.
Step 2, check: if \(\sin x\) or \(\cos x\) equals a number bigger than 1 or less than \(-1\), there are no solutions.
Step 3, first angle: use \(\sin^{-1}\), \(\cos^{-1}\) or \(\tan^{-1}\) on your calculator.
Step 4, the rest: sketch the graph, draw \(y = k\), and use the symmetry to find every crossing in \(0^\circ\) to \(360^\circ\).
Worked example Solving a sine equation
Solve \(\sin x = \dfrac{2}{3}\) for \(0^\circ \le x \le 360^\circ\).
First angle: \(x = \sin^{-1}\left(\dfrac{2}{3}\right) = 41.81\ldots^\circ\).
The sine graph is symmetrical about \(x = 90^\circ\), so the second solution is \(180^\circ - 41.81^\circ = 138.19^\circ\).
Sine is negative from \(180^\circ\) to \(360^\circ\), so there are no more solutions.
Answer: \(x = 41.8^\circ\) or \(x = 138.2^\circ\)
Worked example Rearrange first
Solve \(2\cos x + 1 = 0\) for \(0^\circ \le x \le 360^\circ\).
Subtract 1 and divide by 2: \(\cos x = -\dfrac{1}{2}\).
First angle: \(\cos^{-1}(-0.5) = 120^\circ\).
The cosine graph is symmetrical about \(x = 180^\circ\): \(360^\circ - 120^\circ = 240^\circ\).
Answer: \(x = 120^\circ\) or \(x = 240^\circ\)
Examiner trap
Stopping after the calculator answer. 'Solve for \(0^\circ \le x \le 360^\circ\)' almost always has two answers, and the second one is a mark on its own.
Section 02
Negative values and tangent.
For a negative value of \(\sin x\) or \(\tan x\), the calculator gives a negative angle, which is outside the range. Use it to find the angles that ARE in the range, then discard it.
Worked example Tangent with a negative value
Solve \(\tan x = -2\) for \(0^\circ \le x \le 360^\circ\).
Calculator: \(\tan^{-1}(-2) = -63.43\ldots^\circ\), which is outside the range.
The tangent graph repeats every \(180^\circ\): \(-63.43^\circ + 180^\circ = 116.57^\circ\).
Add another \(180^\circ\): \(116.57^\circ + 180^\circ = 296.57^\circ\).
Answer: \(x = 116.6^\circ\) or \(x = 296.6^\circ\)
Worked example Sine with a negative value
Solve \(5\sin x + 3 = 1\) for \(0^\circ \le x \le 360^\circ\).
Rearrange: \(5\sin x = -2\), so \(\sin x = -0.4\).
Calculator: \(\sin^{-1}(-0.4) = -23.58\ldots^\circ\). The acute angle with \(\sin = 0.4\) is \(23.58^\circ\).
Sine is negative between \(180^\circ\) and \(360^\circ\): \(180^\circ + 23.58^\circ = 203.58^\circ\) and \(360^\circ - 23.58^\circ = 336.42^\circ\).
Answer: \(x = 203.6^\circ\) or \(x = 336.4^\circ\)
Your turn
Solve \(4\cos x - 1 = 0\) for \(0^\circ \le x \le 360^\circ\).
\(\cos x = 0.25\), so \(x = \cos^{-1}(0.25) = 75.5^\circ\), and \(360^\circ - 75.5^\circ = 284.5^\circ\).
Examiner trap
Giving \(-63.4^\circ\) as an answer. It solves the equation but is outside \(0^\circ \le x \le 360^\circ\), so it earns nothing. Use it only as a stepping stone.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Your calculator gives \(\sin^{-1}(0.3) = 17.5^\circ\). What is the other solution of \(\sin x = 0.3\) for \(0^\circ \le x \le 360^\circ\)?
Quick check+10 XP
Solve \(\cos x = 0.8\) for \(0^\circ \le x \le 360^\circ\).
Quick check+10 XP
Solve \(\tan x = 1\) for \(0^\circ \le x \le 360^\circ\).
Quick check+10 XP
How many solutions does \(\sin x = -1\) have for \(0^\circ \le x \le 360^\circ\)?
Section 03
No solutions, and exact solutions.
Sine and cosine can never be bigger than 1 or less than \(-1\), so some equations have no solutions at all. When the value is one of the exact values, solve without a calculator.
Worked example An impossible equation
Solve \(3\sin x + 5 = 1\) for \(0^\circ \le x \le 360^\circ\).
Rearrange: \(3\sin x = -4\), so \(\sin x = -\dfrac{4}{3} = -1.33\ldots\)
\(-1.33\ldots < -1\), and \(\sin x\) is never less than \(-1\). (The calculator shows Math ERROR.)
Answer: No solutions
Worked example Exact solutions, without a calculator
Without a calculator, solve \(2\sin x = \sqrt{3}\) for \(0^\circ \le x \le 360^\circ\).
\(\sin x = \dfrac{\sqrt{3}}{2}\), and \(\sin 60^\circ = \dfrac{\sqrt{3}}{2}\).
Second solution: \(180^\circ - 60^\circ = 120^\circ\).
Answer: \(x = 60^\circ\) or \(x = 120^\circ\)
Examiner trap
Using the pattern for the wrong function. For \(\cos x = 0.25\), the partner of \(75.5^\circ\) is \(360^\circ - 75.5^\circ\), not \(180^\circ - 75.5^\circ\). A quick sketch of the right graph prevents this.
Stage 2 · Exam
Exam-style questions
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Exam style+20 XP
Solve \(4\sin x - 1 = 2\) for \(0^\circ \le x \le 360^\circ\).
Exam style+20 XP
Solve \(5\cos x + 2 = 0\) for \(0^\circ \le x \le 360^\circ\).
Exam style+20 XP
Solve \(2\tan x + 1 = 0\) for \(0^\circ \le x \le 360^\circ\).
Exam style+20 XP
How many solutions does \(3\sin x + 5 = 1\) have for \(0^\circ \le x \le 360^\circ\)?
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