Solving equations in a range
To solve an equation such as sin x = k for 0° ≤ x ≤ 360°, first find the acute basic angle from the positive value of k. Then use the CAST rule to place solutions in the correct quadrants. Every basic angle gives more than one solution because the function repeats.
Key formula /
If sin x = k (k > 0): solutions are α and 180° − α. If cos x = k (k > 0): α and 360° − α. If tan x = k (k > 0): α and 180° + α. For a negative k, keep the basic angle from |k| and use the negative quadrants.
Modelling with trig functions
Many real quantities — tides, the height of a Ferris-wheel seat, temperature — vary in a wave pattern modelled by h = a sin(bt) + c or h = a cos(bt) + c. The maximum is c + |a| and the minimum is c − |a|.
When the equation involves a multiple angle such as sin 2x = k, first widen the range so it covers 2x, solve for 2x, then divide each answer by the multiplier. This is why multiple-angle equations usually have more solutions than the simple version. A tidy habit is to state the basic angle once, generate every candidate solution from it using CAST, and only at the end reject any that fall outside the required range.
Worked example
Solve 2 sin x = 1 for 0° ≤ x ≤ 360°. Then sin x = 0.5, basic angle 30°. Sine is positive in quadrants 1 and 2, so x = 30° and x = 180° − 30° = 150°.
A seat height is h = 6 + 4 sin θ metres. The greatest height is 6 + 4 = 10 m.
Remember
- Find the basic (acute) angle first.
- Use CAST to decide which quadrants give solutions.
- Check every solution lies inside the stated range.