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Find the exact value of sin⁻¹(sin 5π/6) + tan⁻¹ 1.

1. tan⁻¹ 1 = π/4✓ 1
2. Exact value, inside the range of tan⁻¹
3. sin⁻¹(sin 5π/6) = 5π/60 / 1
4. Marker: sin⁻¹ only returns values in [−π/2, π/2]. sin 5π/6 = ½, so sin⁻¹(½) = π/6 and the total is 5π/12.
5. ∴ total = 13π/12

Evaluate ∫ from 0 to 2π of √(1 − cos 2x) dx.

1. 1 − cos 2x = 2 sin²x✓ 1
2. The right double-angle form for this root
3. √(2 sin²x) = √2 sin x0 / 1
4. Marker: √(sin²x) = |sin x|. For π < x < 2π, sin x is negative, so the integral splits.
5. √2 [−cos x] from 0 to 2π = 00 / 1
6. Marker: A positive integrand cannot give 0. The value is 4√2.