Example 5: Arc Sine and Arc Cosine

Show that arcsin y + arccos y = π/2 (Figure 7.3.6). Let θ = arcsin y. We have y = sin θ = cos (π/2 - θ). Also, when - π/2 ≤ θ ≤ π/2, we have

π/2 ≥ - θ ≥ -π/2, π ≥ π/2 - θ ≥ 0.

Thus

π/2 - θ = arccos y,

arcsin y + arccos y = θ + (π/2 - θ) = π/2.

Figure 7.3.6