Download Section 17.1 - Degrees and Radians

Section 17.1 – Angles of rotation and radian measure.
When first introduced to trigonometry in geometry class we only considered the trig functions
with respect to a right triangle. Is it possible to find the value of the sine of 90° or some larger
angle? It is if we think of an angle created by a fixed ray and a ray that rotates about an origin. If
we think of this fixed ray as the positive x-axis we can rotate the other ray around the coordinate
plane through the other quadrants.
The most common way to measure angles is in degrees which is just
of a full circle rotation.
Degrees are fine but they have limitations. Since a degree has no concrete relationship to any other linear measure another type of angle measure
is needed to solve this problem. A radian solves this problem and is the type of angle measure used at the calculus level and above.
Think of radian measure has highly related to arc-length. Most radian measures
are in terms of π since the arc-length of a circle (circumference) is C=2πr.
What are each of the following rotations in degrees and radians?
1)Full rotation 2)Half rotation 3)quarter rotation 4) 1/3 of a rotation
5)1/6 of a rotation 6)1/8 of a rotation 7)1/12 of a rotation
We can convert from degrees to radians by multiplying by
We can convert from radians to degrees by multiplying by
°
°
While there is nothing wrong with these rules, to convert from radians to degrees it is usually easier just to substitute 180 for π.
Write each radian measure in degrees and each degree measure in radians then sketch in standard position. (Angles given in terms of π are usually
assumed to be radians.)8) 240° 9)135° 10)-330°(this is a clockwise rotation) 11)
12) -
13) 1 radian.
Some angles are conterminal. This means that the terminal side ends up pointing in the same place. For example 40° and 400° are conterminal
angles and are conterminal angles. We can find conterminal angles by adding or subtracting multiples of 360°(in degrees) or 2π(in
radians to a given angle. Find the nearest two positive and two negative conterminal angles for each and demonstrate the original angle and one
co-terminal angle with a sketch. 14) 70° 15)−
16)
17) - 500°
Since a radian measure is the quotient of the arc-length and the radius we can solve for the arc-length s and get s=rθ. This is the formula for arc
length given a radius and a central angle θ in radians(not degrees). Find the arc length of each:
18) A sector with radius 4in and angle measure of
far does the tip travel in feet in one swing?
19) The tip of a 4ft pendulum of a grandfather clock in one swing makes an arc of 70°. How