Convert between degrees and radians, understand reference angles. Math 30-1 Alberta.
Lesson 4.1 of Trigonometric Ratios & Functions in Math 30-1 — Alberta curriculum lessons.
Definition. You already know how to measure angles in degrees (like 90° or 180°).
A radian is another way to measure angles — but instead of being based on a scale like degrees, it's based on the circle itself.
The Simple Idea. Imagine a circle with radius r.
Now imagine taking a piece of string that is exactly the same length as the radius.
👉 Wrap that string along the edge of the circle.
The angle formed at the center is 1 radian.
Key Relationship. When: a = r
Then: θ = 1 radian
👉 That's the definition of a radian
The Most Important Formula. From this idea, we get:
a = rθ
This is the arc length formula.
You can rearrange it:
θ = (a)/(r)
r = (a)/(θ)
What each variable means:
θ = angle (in radians)
a = arc length (curved distance along the circle)
r = radius
Full Rotation. A full rotation corresponds to one complete revolution around a circle.
In degrees:. Full rotation = 360°
Radians and Arc Length. Recall the definition of a radian:
θ = (a)/(r)
where:
a = arc length
r = radius
Applying This to a Full Circle. For a full rotation, the arc length is the circumference of the circle: a = 2π r. Substitute into the radian formula:
θ = (a)/(r) = (2π r)/(r) = 2π
Key Result. 360° = 2π radians
This is the fundamental relationship between degrees and radians.
The Unit Circle. We use a unit circle (radius = 1) because when r = 1, the formula a = rθ becomes a = θ. This means the arc length equals the angle in radians, making calculations much simpler.
Example 1: Finding the Radius. An arc of 18.5 cm has a central angle of (π)/(3) radians. Determine the radius of the circle, to the nearest tenth of a cm.
Solution. Given: a = 18.5 cm, θ = (π)/(3) radians
Find: r (radius)
We know: θ = (a)/(r), so r = (a)/(θ)
r = (18.5)/((π)/(3)) = (18.5 × 3)/(π) = (55.5)/(π) = 17.7 cm
Answer: The radius is 17.7 cm.
Example 2: Finding the Angle. A pendulum is 25 cm long. If the end of the pendulum swings through a total distance of 8.3 cm, what is the measure, in radians and in degrees (both to the nearest tenth), of the angle through which the pendulum swings?
Solution. Given: r = 25 cm, a = 8.3 cm
Find: θ (in radians and degrees)
We know: θ = (a)/(r)
θ = (8.3)/(25) = 0.3 rad
0.3 × (180)/(π) = (54)/(π) ≈ 17.2°
Answer: The angle is 0.3 rad or 17.2°.
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