Radians – AS Maths Revision – Core Mathematics (C2)

By The Way (before we start)

  • You should know how to use the sine and cosine rules, from GCSE Maths
  • The cosine rule is given in the formula book in Core 2, so there’s no need to memorize it (you can rearrange it yourself if necessary)
  • The sine rule is “a over sin A equals b over sin B equals c over sin C”
  • Also the other way up, as in “sin A over a equals sin B over b equals sin C over c”
  • The sine rule is useful when you know an angle opposite a side, and want to find out either an angle, or a side which you know the side or angle (respectively) opposite
  • The cosine rule can allow you to work out a side using a known angle between two sides, or it can let you work out an angle, if you know all three sides
  • Rather than remembering alternate forms of the cosine rule, just re-label the triangle
  • Oh, and I guess you still know the old SOH CAH TOA trigonometry stuff

Where were we?

  • Radians are an alternative way of measuring angles
  • In fact, they’re better than degrees, because they’re more fundamental / constant, less arbitrary / contrived
  • If you have an arc that’s of the same length as the radius of the circle, and you draw lines from both ends of the arc to the centre of the circle, the angle they meet at will be 1 radian
  • In other words, 1 radian is the angle at the centre of a circle, subtended by the ends of an arc of the same length as the radius
  • Angles in radians tend to be written in terms of Pi, to avoid irrational numbers and rounding
  • To convert between radians and degrees, multiply or divide by Pi, and divide or multiply by 180
  • Angles expressed in degrees will have a bigger number than angles expressed in radians, because there are more degrees in a circle than radians
  • Thus, to convert from degrees to radians, divide by 180 and multiply by Pi
  • To convert from radians to degrees, multiply by 180 and divide by Pi

Arcs, Sectors, Chords and Segments

  • I just need to define some terms for bits of circles, before the next part
  • An arc is a bit of the circumference – the line around the edge of the circle, between two points
  • You can have a major arc and a minor arc between two points on the circumference (the minor arc takes the short route, and the major arc takes the long way round)
  • If you were to just draw a straight line from two points on the circumference, it would cut into the circle and make it look liked you chipped some off of it
  • The straight line itself would be called a cord, and the chipped-off area it created would be called a segment
  • A sector is a part of the circle – on a pie chart, different sectors represent different things. Oh, and you normally cut pizzas/cakes into sectors – isosceles triangles, but with the odd-one-out side curved

Uses

Definition of an Angle in terms of radius and ...

Definition of an Angle in terms of radius and arc length. (Photo credit: Wikipedia)

 

  • There are some good reasons for you to adopt this new way of measuring angles…
  • All the trig stuff that works in degrees seems to work in radians
  • In addition, here are some useful things that only work in radians…
  • (If you were desperate, you could convert degrees to radians and back again at the end, but you might as well just work in radians the whole time, unless it’s a degrees question)
  • l=r*θ, where l is the arc length, r is the radius of the circle and θ, theta, is the angle (IN RADIANS)
  • Area of Sector = 0.5 * r^2 * θ [“sector area equals half r squared theta”], where r is the radius, and theta is the angle in the sector (IN RADIANS)

WARNING

  • The problem is… sin, cos and tan work differently for radians, so before you ever sin, cos, tan, sin^-1, cos^-1 or tan^-1 in Core 2, you must make sure your calculator is in the correct mode!
  • It’ll have a degrees mode and a radians mode (and maybe some other mode, but goodness knows what that is…) JUST MAKE SURE YOU’RE IN THE RIGHT MODE FOR EACH QUESTION (oops. Caps.)
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About Matt

I like writing, filmmaking, programming and gaming, and prefer creating media to consuming it. On the topic of consumption, I'm also a big fan of eating.
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