The general solution of sin α = cos β, sec α = csc β and tan α = cot β
Because sin α = cos(90° − α), the equation sin α = cos β says that β and 90° − α have equal cosines. For acute angles that means β = 90° − α, so α + β = 90°. The same reasoning gives α + β = 90° for sec α = csc β and for tan α = cot β. If α and β are not both acute, other solutions appear because cos is also equal at angles on the other side of the x-axis and every 360° (every 180° for tan). The table shows the general solution. In an exercise solve the equation α + β = 90° first, then check the answer by substituting, and keep only the solutions inside the stated interval.
| Equation | Acute angles | In general (n a whole number) |
|---|---|---|
| sin α = cos β | α + β = 90° | α + β = 90° + 360°n or α − β = 90° + 360°n |
| sec α = csc β | α + β = 90° | α + β = 90° + 360°n or β − α = 90° + 360°n |
| tan α = cot β | α + β = 90° | α + β = 90° + 180°n |
Pairs that add to 90. Write 6 pairs of acute angles whose sum is 90°. For each pair check with a calculator that sin of one equals cos of the other and tan of one equals cot of the other.