13.2 Define General Angles and Use Radian Measure
Name ______________________________________________ Date ______________
In right triangle trig, the angles that we use are all acute, and even if we consider other
kinds of triangles, angle measures will always be less than 180. Over the next couple
days, however, we will learn how angle measures and trig functions can be extended to
measures over 180 and even negative measures.
Part 1 – Angles as Rotations
To do this, we shift from thinking of angles as a corner of a polygon to thinking of them as
rotations. Here are the general guidelines:
 Unless otherwise specified, angles will be assumed to be in standard position, i.e. with the
initial side along the positive x-axis.
 A counterclockwise rotation corresponds to a positive angle.
 A clockwise rotation corresponds to a negative angle.
 An angle’s quadrant is determined by the quadrant of its terminal side.
The diagram below shows what this looks like.
1. Represent each angle as a rotation in standard position.
(a) 150
(b) 315
(c) -120
2. Represent the angles 90, 450 and -270 as rotations in standard position. What do you
observe?
Angles with the same terminal side are called coterminal angles. You can find coterminal
angles by adding or subtracting multiples of 360.
3. Find one positive angle and one negative angle that is coterminal with the given angle.
(a) 315
(b) -120
4. Write a formula for the measures of all angles coterminal with 150.
It’s time for a brief look at how this works with trig. We’ll do a lot more of this in the
next couple lessons.
5. Draw an angle  in standard position whose terminal side passes through P(8, 6).
Then find the values of the six trigonometric functions of . [Can you see how to
draw in a right triangle? The terminal side becomes the hypotenuse.]
Part 2 – Radian Measure
Angles can be measured in degrees or in another kind of unit called radians. As we will
see in Chapter 14, radians are generally used in non-geometric applications of trig.
Here is a definition of a radian. First study the circle centered at the origin with radius r.
An angle of one radian is the measure of q in standard position whose terminal side
intercepts an arc of length r. (Think of it as the angle of a pizza wedge that is created by
two congruent sides of length r where the length of the crust is also r…sort of like an
equi-“lateral” piece of pizza.) This gives a conversion of 1 radian » 57.3 .
Because the circumference of a circle is 2p r , there are 2p radians in
a full circle. Degree measure and radian measure are therefore related
by the equations 360 = 2p radians, or 180 = p radians. This
conversion based on  is much more commonly used than the 57.3
one noted above.
Here are the steps to follow to convert between degrees and radians:
To Convert Degrees to Radians:
Multiply degree measure by
p radians
180
To Convert Radians to Degrees:
Multiply radian measure by
180
p radians
6. Convert to radians.
(a) 210
(b) 270
7. Convert to degrees.
7p
(a)
radians
9
(b)
-
p
12
The diagram below shows a unit circle (radius = 1 unit)
showing radian measures for the multiples of the special
angles 30 , 45 , 60 , 90 . Write the corresponding degree
measure next to each radian measure the unit circle.
8. Evaluate the trig functions. Can you do this without a calculator?
(a) cos
p
3
(b) sin
p
4
(c) tan
p
6
Part 3 – Arc length and area of a sector
Do you recall from geometry what a sector is? A sector is a region of a circle that is
bounded by _______ _______________ and an arc of the circle.
9. Consider a circle of radius 10 cm, and a sector of this circle whose central angle is
100. Make a diagram, then find the arc length and area of the sector.
Write formulas for the arc length and area of a sector for a central angle  measured in
degrees.
There are different versions of these two formulas if q is measured in radians.
Arc Length of a Sector:
s = r ×q
Area of a Sector:
A=
1
2
r ×q
2
10. Find the arc length and the area of a sector with a central angle of
2p
and a radius of
5
25 cm.
11. A scientist performed an experiment to study the effects of gravitational force on humans.
In order for humans to experience twice Earth’s gravity, they were placed in a centrifuge
58 feet long (that is, the diameter is 58 ft) and spun at a rate of about 15 revolutions per
minute.
(a) Through how many radians did the people rotate each second?
(b) Find the length of the arc through which the people rotated each second.