Equations with scaled and shifted angles

Equations with scaled and shifted angles

Practise trigonometric equations whose angle is a linear expression in the unknown. Scaling or shifting an angle changes both the solution spacing and the interval that must be searched. This set emphasizes complete solution sets for positive, negative and fractional multipliers rather than assuming every angle is a positive whole-number multiple.

Single-select and multiselect questions cover interval mapping, shifted sine and cosine equations, tangent families, fractional arguments and endpoint checks. Error-analysis tasks show how searching only one cycle of an auxiliary angle can omit valid answers. A constant-angle case also tests whether a division step is actually allowed.

A reliable workflow

Review elementary trigonometric equations and affine transformations. Introduce a new variable for the entire angle, transform both endpoints of the original interval, solve in that transformed interval and then convert every solution back. A negative coefficient reverses order and moves open and closed endpoints with their original values. Finally, substitute into the original equation and check the original interval. Worked explanations show each stage and keep radians explicit throughout.