Sine — Geometry, what it means and where it is taught
sin θ = oppositehypotenuse.
Taught in Geometry.
Where it is taught
What it rests on
Work through these first if this one does not settle.
- Hypotenuse — The side opposite the right angle — always the longest.
What uses it later
This idea is needed for:
- Arcsine — The inverse of sine, restricted to outputs from -90° to 90°.
Practise this idea
15 questions in the course bank use Sine, in Right Triangle Trigonometry, Radians & the Unit Circle, Graphs of Sine & Cosine, Trigonometric Identities & Equations, Inverse Trigonometric Functions, The Law of Sines & Cosines. The first 8 are below; the rest are in those units.
Problem 1
A ladder 6 m long leans at 70°. How high does it reach? Round to one decimal place.
Worked answer
5.6
sin 70° = h6
h = 6 × 0.9397 ≈ 5.6 m
Problem 2
sin-1(0.5) means
- the angle whose sine is 0.5
- 1sin 0.5
- sin(0.5) doubled
- half of sin
Worked answer
the angle whose sine is 0.5
It undoes the sine.
sin-1(0.5) = 30°.
Problem 3
In quadrant III, which is true?
- Both sine and cosine are negative
- Sine positive, cosine negative
- Both positive
- Sine negative, cosine positive
Worked answer
Both sine and cosine are negative
Quadrant III has x<0 and y<0
cos follows x, sin follows y
Problem 4
A sine wave has period 6. Find b in y=sin(bx). Give the fraction multiplying π.
Worked answer
1/3
b=2π6=π3
Problem 5
If sinθ=14, find cscθ.
Worked answer
4
11/4=4
Problem 6
Solve sinθ=12 for 0° ≤ θ < 360°. Give both answers, smallest first.
Worked answer
30,150
Reference angle 30°
30° and 180-30=150°
Problem 7
How many solutions does sinθ=0.4 have for 0° ≤ θ < 360°?
Worked answer
2
One in quadrant I, one in quadrant II
Problem 8
Solve 2sinθ-1=0 for 0° ≤ θ < 360°. Give both, smallest first.
Worked answer
30,150
sinθ=12
30° and 150°