Finding every solution in the interval

Mathematics · Geometry · ages 16-17

Name ______________________ Date ____________

How many solutions does sin x = 0.5 have for 0 ≤ x ≤ 2π?
Answer: ______________
Solve sin x = 0.5 for 0 ≤ x ≤ 2π.
- π/6 and 5π/6
- π/6 only
- π/3 and 2π/3
- 0 and π
Which pair solves sin x equals 0.5 for x between 0 and 2 pi?
- Pi over 6 only
- Pi over 3 and 2 pi over 3
- Pi over 6 and 5 pi over 6
The equation sin x equals 0.5 has exactly one solution between 0 and 2 pi.
Circle one: True False
2 sin x = 1. What is the smallest non-negative solution in degrees?
Answer: ______________
Dividing an equation by sin x is a safe shortcut.
Circle one: True False
Solve sin 2x = 0.5 for 0 ≤ x ≤ 2π. How many solutions?
- 2
- 8
- 1
- 4
Maya says an equation mixing sin and cos should first become one function. Is Maya right?
Circle one: True False
You hold 2 sin squared x plus sin x minus 1 equals 0. What next?
- Divide by sin x at once
- Read the answers off the constants
- Factorise it like a quadratic in sin x
How many solutions does sin 3x = 0 have for 0 ≤ x ≤ 2π?
Answer: ______________

Answer key

For grown-ups. Fold this page away before handing over the rest.

2 · The twin pair: 2.
π/6 and 5π/6 · Reference π/6 plus its mirror 5π/6. The π/6 only option drops the twin.
Pi over 6 and 5 pi over 6 · The wave hits 0.5 twice, mirrored across pi over 2.
False · It has two solutions, the reference angle and its mirror.
30 · sin x = 0.5 gives 30° first.
False · Division deletes every solution where sin x is zero.
4 · 2x spans two turns with 4 hits. The 2 option solves one turn only.
True · Single function unlocks factorising.
Factorise it like a quadratic in sin x · Treat sin x as the variable and split into two factors.
7 · 3x spans 0 to 6π: hits at 0, π, ..., 6π, seven values.