MathematicsChapter 6: Trigonometry and Bearing
Radians and Conversion
Circular measure uses radians.
Learning objectives
- Explain radians and conversion in clear mathematical language
- Apply the ideas in what a radian is
- Solve problems involving converting
Definition
Circular measure uses radians. The radian measure of a central angle is theta = arc/radius. You get 1 radian when the arc length equals the radius. A full turn has arc 2 pi r, so 2 pi rad = 360 deg. Half of that is pi rad = 180 deg. To convert, rad to deg multiply by 180/pi, and deg to rad multiply by pi/180. Then 45 deg = pi/4 and 150 deg = 5pi/6.
Mathematical form
theta = arc/radius. You get 1 radian when the ar
rad = 360 deg. Half of that is pi rad
deg = pi/4 and 150 deg
rad = 180 deg
Where / symbols
- theta = arc/radius
- The radian measure of a central angle is theta = arc/radius.
- 1 radian
- You get 1 radian when the arc length equals the radius.
- 2 pi rad = 360 deg
- A full turn has arc 2 pi r, so 2 pi rad = 360 deg.
- pi rad = 180 deg
- Half of that is pi rad = 180 deg.
Relationship
- theta = arc/radius
- 2 pi rad = 360 deg
- pi rad = 180 deg
- 45 deg = pi/4
Overview
One full turn, two unit systems:
What A Radian Is
On a circle, theta = arc/radius. If the arc is as long as the radius, theta is 1 radian. The radius unit cancels, so a radian is a pure number, but we still write rad to show the system. A full turn wraps an arc of 2 pi r, hence 2 pi rad = 360 deg. Cutting the turn in half gives pi rad = 180 deg, the conversion anchor you should memorise.
Examples arc = radius gives 1 radian · full turn 2 pi rad = 360 deg · straight angle pi rad = 180 deg
Converting
Rad to deg: multiply by 180/pi. Deg to rad: multiply by pi/180. Worked: 45 deg = 45 * pi/180 = pi/4, so 45 deg = pi/4. Worked: 150 deg = 150 * pi/180 = 5pi/6, so 150 deg = 5pi/6. Reverse: pi/6 rad = 30 deg because (pi/6)*(180/pi)=30. Cancel pi when it appears on top and bottom. Keep the factor 180 next to pi so you do not invert the rule.
Examples 45 deg = pi/4 · 150 deg = 5pi/6 · pi/3 rad = 60 deg
What it means
Circular measure uses radians.
Main ideas
1. WHAT A RADIAN IS: On a circle, theta = arc/radius. 2. CONVERTING: Rad to deg: multiply by 180/pi.
Why it matters
Because pi rad = 180 deg, a degree is pi/180 of a radian. Do not treat 1 rad as 1 deg; 1 rad is about 57 deg. Leave exact answers in terms of pi unless the question asks for a decimal.
How to answer in an exam
The easy memory is pi rad = 180 deg. From that, rad to deg 180/pi and deg to rad pi/180. Drill 45 deg = pi/4 and 150 deg = 5pi/6 until they are automatic.
Explanation
Because pi rad = 180 deg, a degree is pi/180 of a radian. Do not treat 1 rad as 1 deg; 1 rad is about 57 deg. Leave exact answers in terms of pi unless the question asks for a decimal.
In simple words
The easy memory is pi rad = 180 deg. From that, rad to deg 180/pi and deg to rad pi/180. Drill 45 deg = pi/4 and 150 deg = 5pi/6 until they are automatic.
Key points
- theta = arc/radius
- 1 radian
- 2 pi rad = 360 deg
- pi rad = 180 deg
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Radians and Conversion
Sticky Notes · Quick Revision
Core Idea
- Circular measure uses radians.
Key Terms
- theta = arc/radius: The radian measure of a central angle is theta = arc/radius.
- 1 radian: You get 1 radian when the arc length equals the radius.
Main Pattern
- 1. WHAT A RADIAN IS: On a circle, theta = arc/radius.
- 2. CONVERTING: Rad to deg: multiply by 180/pi.
Worked Reminder
- arc = radius gives 1 radian
- full turn 2 pi rad = 360 deg
- straight angle pi rad = 180 deg
Common Mistake
- The easy memory is pi rad = 180 deg. From that, rad to deg 180/pi and deg to rad pi/180. Drill 45 deg = pi/4 and 150 deg = 5pi/6 until they are automatic.
Exam Tip
- State the rule for radians and conversion, show each algebraic step, and check the final answer against the question.
60-second revision
- Circular measure uses radians.
- On a circle, theta = arc/radius.
- Rad to deg: multiply by 180/pi.
- Because pi rad = 180 deg, a degree is pi/180 of a radian.
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