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3.10 · Eq vs Inverse
sin(x)=½ vs sin⁻¹(½)=x. Are they equivalent? How many solutions does each have?
Periodic vs restricted domain
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3.10 · General Solution
Write the general solutions for sin(x)=½, cos(x)=½, and tan(x)=1.
How many families for each? What period added?
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3.10 · tan General Soln
Why does tan(x)=1 need only ONE general solution family (not two)?
Think about tan's period
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3.10 · Ex1 General
Find all solutions to cos(x) = −√2/2.
Reference angle: π/4. Negative cos → Q2, Q3.
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3.10 · Ex3 General
Find all solutions to tan(x) = 1/√3.
tan's period = π → one family
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3.10 · Ex5 Intersection
f(θ)=3+2cosθ, g(θ)=2. Find θ in [0,2π) where they intersect.
Set f=g, isolate cos
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3.10 · Ex6 Quadratic
Find zeros of g(θ)=3−4sin²θ on [0,2π).
sin²θ = 3/4 → sinθ = ±√3/2
4 solutions
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3.10 · Ex7 Factor
Find all θ in [0,2π) where cos²θ + cosθ = 0.
Factor: cosθ(cosθ+1)=0
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✓ General Solutions
sin(x)=½: x=π/6+2πk and x=5π/6+2πk · cos(x)=½: x=π/3+2πk and x=5π/3+2πk · tan(x)=1: x=π/4+πk
1/10
✓ Eq vs Inverse Trig
NOT equivalent. sin(x)=½ has infinitely many solutions (π/6+2πk and 5π/6+2πk). sin⁻¹(½)=π/6 only (restricted to [−π/2,π/2]).
4/10
✓ Ex1 — General Solution
x = 3π/4 + 2πk and x = 5π/4 + 2πk
Ref angle π/4. cos negative → Q2 (3π/4) and Q3 (5π/4). Add 2πk to each.
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✓ tan — One Family
tan period=π. Solutions π/4 and 5π/4 are exactly π apart, so x=π/4+πk captures both. No second family needed.
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✓ Ex5 — Intersections
θ = 2π/3 and θ = 4π/3
3+2cosθ=2 → cosθ=−1/2 → Q2: 2π/3, Q3: 4π/3.
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✓ Ex3 — tan General
x = π/6 + πk
tan=1/√3 at π/6 (Q1). tan period=π → x=π/6+πk covers Q1 and Q3 solutions.
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✓ Ex7 — Factor
θ = π/2, π, 3π/2
cosθ(cosθ+1)=0 → cosθ=0 (π/2, 3π/2), cosθ=−1 (π).
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✓ Ex6 — Quadratic
θ = π/3, 2π/3, 4π/3, 5π/3
sin²θ=3/4 → sinθ=±√3/2. +√3/2: π/3, 2π/3. −√3/2: 4π/3, 5π/3.
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3.10 · Ineq Circle
Find all θ in [0,2π) where sinθ ≥ √2/2.
Which arc of unit circle has y ≥ √2/2?
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3.10 · Sign Chart Ineq
Find all θ in [0,2π] where 2sin²θ + sinθ < 0.
Factor: sinθ(2sinθ+1)=0
Zeros: 0, π, 7π/6, 11π/6, 2π
10/10
✓ Sign Chart — Ex5
π < θ < 7π/6 OR 11π/6 < θ < 2π
sinθ(2sinθ+1)=0: zeros at 0,π,7π/6,11π/6,2π. Test intervals: product negative in (π,7π/6) and (11π/6,2π).
9/10
✓ Ineq — Unit Circle
π/4 ≤ θ ≤ 3π/4
sinθ=√2/2 at π/4 and 3π/4. sinθ≥√2/2 = arc from π/4 to 3π/4. Include endpoints (≥).