Learn the method, study an example, then explain your reasoning in an exam-style problem. Work on paper before opening the solution and compare each step with the marking guidance.
Trigonometric terms in an arithmetic progression
Translate equal differences into an equation and recover every permitted angle.
Notes
Before you start: use sin²x+cos²x=1, exact values, quadrants and radian intervals.
Three real numbers u,v,w are consecutive AP terms exactly when v−u=w−v, or \(2v=u+w\). This works even when a term or the common difference is zero.
Substitute the given trigonometric expressions into this condition before simplifying. Retain any stated angle interval. If division by sin x or cos x is helpful, test its zero cases in the original equation first.
After solving for an angle, use d=v−u to find the common difference. The AP condition by itself does not mean the terms are positive or increasing. A principal inverse value can miss another branch.
Recall later
For consecutive terms sin²x,k,cos²x, the AP condition gives 2k=1, so k=1/2 independently of x.
Common mistakes
Setting a product condition instead of equal differences for an AP.
Dividing by cosine without checking cosine-zero cases.
Assuming the difference is positive or omitting a second quadrant branch.
Three worked examples
Example 1
Show that cos x, 0, −cos x are consecutive AP terms for every real x.
Show the reasoning
The differences are both −cos x. Equivalently \(2(0)=\cos x-\cos x\). At cosine-zero angles all three terms and the difference are zero, which is still an AP.
Example 2
Find x in 0≤x<2π when sin x, cos x, 0 are consecutive AP terms.
Show the reasoning
The condition is \(2\cos x=\sin x\). If cos x=0, sine is ±1 and the equation fails. Otherwise tan x=2. Hence \(x=\arctan(2)\) or \(\pi+\arctan(2)\), approximately 1.11 and 4.25 rad to 3 significant figures.
Example 3
For consecutive AP terms sin²x,k,cos²x, find k and d.
Show the reasoning
\(2k=\sin^2x+\cos^2x=1\) gives k=1/2. Then \(d=1/2-\sin^2x=\cos^2x-1/2\). Its sign depends on the angle.
