Cosine — Geometry, what it means and where it is taught
cos θ = adjacenthypotenuse.
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.
- Adjacent — The side next to that angle which is not the hypotenuse.
What uses it later
This idea is needed for:
- Arccosine — The inverse of cosine, restricted to outputs from 0° to 180°.
Practise this idea
8 questions in the course bank use Cosine, in Radians & the Unit Circle, Trigonometric Identities & Equations, Inverse Trigonometric Functions, Parametric Equations.
Problem 1
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 2
If cosθ=12, find secθ.
Worked answer
2
11/2=2
Problem 3
Solve cosθ=12 for 0° ≤ θ < 360°. Give both, smallest first.
Worked answer
60,300
Reference angle 60°
60° and 360-60=300°
Problem 4
Find arccos 0 in degrees.
Worked answer
90
cos 90° = 0
Problem 5
Find arccos 1 in degrees.
Worked answer
0
cos 0° = 1
Problem 6
Find arccos(-12) in degrees.
Worked answer
120
Reference angle 60°
Cosine is negative in quadrant II, so 180-60=120
Problem 7
What is the range of arccos?
- 0° to 180°
- -90° to 90°
- 0° to 90°
- -180° to 180°
Worked answer
0° to 180°
Cosine falls from 1 to -1 across 0° to 180°
That covers its whole range exactly once
Problem 8
For x=4cos t, y=3sin t, find the largest value x can take.
Worked answer
4
4(1) = 4