For an equation in u=ax+b, solve for u over the interval obtained from the stated bounds on x, then undo the linear change. A full sine/cosine cycle has length 2pi, while a tangent cycle has length pi. Scaling an argument changes how much of these cycles is covered; it does not require expanding the trigonometric function. 1. Transform both endpoints and their inclusion signs. 2. Find all suitable u-values inside that interval. 3. Recover x=(u-b)/a and check the original bounds. For negative a, reverse the interval order. Example: cos(x/2)=0, 0 <= x 2pi. The argument lies in [0,pi], so x/2=pi/2 and x=pi. Exam check: do not divide the usual list of angles by the multiplier until the complete transformed interval has been considered. Set a calculator to radians when needed.

For an equation in \(u=ax+b\), solve for \(u\) over the interval obtained from the stated bounds on \(x\), then undo the linear change.

A full sine/cosine cycle has length \(2\pi\), while a tangent cycle has length \(\pi\). Scaling an argument changes how much of these cycles is covered; it does not require expanding the trigonometric function.

1. Transform both endpoints and their inclusion signs. 2. Find all suitable \(u\)-values inside that interval. 3. Recover \(x=\frac{u-b}{a}\) and check the original bounds. For negative \(a\), reverse the interval order.

Example: \(\cos\left(\frac{x}{2}\right)=0\), \(0\le x\le2\pi\). The argument lies in \([0,\pi]\), so \(\frac{x}{2}=\frac{\pi}{2}\) and \(x=\pi\).

Exam check: do not divide the usual list of angles by the multiplier until the complete transformed interval has been considered. Set a calculator to radians when needed.

Radian argument intervals

Radian argument intervals