4.1 Radian and Degree Measure
Trigonometry means measurement of triangles.
When placed on the coordinate system, the initial side is placed on the x-axis. This is standard position of the angle. Positive angles are generated by counterclockwise rotation, and negative angles by clockwise rotation. An angle is determined by rotating a ray about its endpoint. The starting position of the ray is the initial side of the angle, and the position after rotation is the terminal side. The endpoint of the ray is the vertex of the angle. In standard position, the vertex of the angle is on the origin or where the x-axis and y-axis intersect (point (0, 0)).
If two angles have the same initial and terminal sides, one clockwise angle and one counterclockwise angle, these angles are coterminal.
Examples:
What is coterminal to:
1. 75 degrees ________
2. 125 degrees ________________
3. π/3 radians ___________
4. 3π/4 radians ______________
Radian Measure - one way to measure angles.
One radian is the measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle.
Recall: the radian measure of an angle of one full revolution is 2π, therefore
½ revolution = π radians
1/4 revolution = π/2 radians
1/6 revolution = π/3 radians
1/8 revolution = π/4 radians
1/12 revolution = π/6 radians
Two positive angles a and b are complementary (complements of each other) if their sum is π/2 or 90 degrees.
Two positive angles a and b are supplementary (supplements of each other) if their sum is π or 180 degrees.
Examples - What is complementary and supplementary to:
1. π/3 Complementary _______________
Supplementary ______________
2. 62 degrees - Complementary __________
Supplementary ______________
Degree Measure:
A measure of one degree is equivalent to a rotation of 1/360 of a complete revolution about the vertex. A full rotation is 360 degrees. Therefore:
½ rotation = 180 degrees
1/4 rotation = 90 degrees
1/6 rotation = 60 degrees
1/8 rotation = 45 degrees
1/12 rotation = 30 degrees
With this in mind:
½ rotation = 180 degrees and ½ rotation = π radians, we can conclude:
½ rotation = 180 degrees = π radians
1/4 rotation = 90 degrees = π/2 radians
1/6 rotation = 60 degrees = π/3 radians
1/8 rotation = 45 degrees = π/4 radians
1/12 rotation = 30 degrees = π/6 radians
1 degree = π/180 radians and 1 radian = 180degrees/π
Quadrants:
I. 0 degrees to 90 degrees OR 0 radians to π/2 radians
II. 90 degrees to 180 degrees OR π/2 radians to π radians
III. 180 degrees to 270 degrees OR π radians to 3π/2 radians
IV. 270 degrees to 360 degrees OR 3π/2 radians to 2π radians
Example: Which quadrant does the angle lie?
1. 72 degrees _________
2. 158 degrees __________
3. -260 degrees ______
To convert degrees to radians, multiply degrees by π radians/ 180 degrees.
To convert radians to degrees, multiply radians by 180 degrees/π radians.
Examples:
1. 83.7 degrees = _______radians
2. -7π/6 radians = ________degrees
3. 395 degrees = _______radians
4. 28π/15 radians = ______degrees
To find the arc length of made by a central angle with a given radius:
S = θ r
Examples:
Given radius 5 cm and central angle of 2π/3, what is the arc length ________
Given radius 8 feet and central angle 35 degrees, what is the arc length _______
Linear and Angular Speed:
Consider a particle moving at a constant speed along a circular arc of radius r. If s is the length of the arc traveled in time t, then the linear speed of the particle is:
Linear speed = (arc length)/time = s/t.
Moreover, if θ is the angle (in radian measure) corresponding to the arc length s, the angular speed of the particle is:
Angular speed = (central angle)/time = θ/t.
Example:
An earth satellite in circular orbit 1250 kilometers high makes one complete revolution every 90 minutes. What is its linear speed? Use 6400 kilometers for the radius of the earth.
HW 32, pg. 291; #7-21 odd, 33, 34, 35-61 odd, 67-73 odd, 75-85 odd, 95
