Angles and Radian Measure Section 4.1

Objectives Estimate the radian measure of an angle shown in a picture. Find a point on the unit circle given one coordinate and the quadrant in which the point lies. Determine the coordinates of a point on the unit circle given a point on the unit circle. Find coterminal angles. Convert angle measures between radians and degrees. Determine the linear speed of an object traveling in a circular motion. Determine the arc length on a given circle.

Vocabulary unit circle radian measure of an angle (radians) degrees vertex of an angle terminal side of an angle initial side of an angle linear speed (length per unit of time) length of a circular arc angular speed (radians per unit of time)

Unit Circle

Consider the picture below. The angle θ is an integer when measured in radians. Give the measure of the angle. The angle that is straight up (right) is approximately radian. The straight angle is approximately radians. Since this angle is closer to the straight angle than to the right angle, the radian measure would be about 3 radians.

Coterminal Angles: Angles are coterminal if they are in standard position and have the same terminal side. Find an angle between 0 and 2π that is coterminalto the angleFind an angle between 0 and 2π that is coterminalto the angle Since this angle is positive, we need to subtract multiples of 2π to find a coterminal angle. To determine how many multiples, we can start by dividing 77 by 3. This will tell us how many half circles there are. Since we need full times around the circle, we need to divide that number by 2 for how many multiples of 2π we need to subtract. Find an angle between 0 and 2π that is coterminal to the angle continued on next slide

Coterminal Angles: Angles are coterminal if they are in standard position and have the same terminal side. Find an angle between 0 and 2π that is coterminalto the angleFind an angle between 0 and 2π that is coterminalto the angle Find an angle between 0 and 2π that is coterminal to the angle 25 full half-ways around a circle full times circles

Conversion between degrees and radian Length of a circular arc Linear speed Angular speed Formulas

Find the degree measure of an angle with radian measure To convert from radians to degrees, we want to multiply the radian measure by a fraction made up of radians and degrees equal to 1. Since we are trying to get rid of the radians, we will need the fraction to have radians in the denominator. Since 180 degrees is the same as π radians, we will use these two numbers in our fraction.

Find the radian measure of an angle with degree measure To convert from degrees to radian, we want to multiply the degree measure by a fraction made up of radians and degrees equal to 1. Since we are trying to get rid of the degrees, we will need the fraction to have degrees in the denominator. Since 180 degrees is the same as π radians, we will use these two numbers in our fraction.

Find the length of the arc on a circle of radius r = 6 inches intercepted by a central angle θ = 135 degrees For this problem, we will use the formula for the length of a circular arc. In order to do this, we must change the angle measure to radians. Now we can plug this into the formula to get

A Ferris wheel has a radius of 30 feet and is rotating at 3.5 revolutions per minute. Find the linear speed, in feet per minute, of a seat on the Ferris wheel. For this we will need the linear speed formula. We will need to calculate s (the length of the circular arc that the Ferris wheel goes through and find the time t that is takes to go through that arc. The Ferris wheel goes through 3.5 revolutions in one minute. This means that the angle is 3.5 times around the circle. Since one time around the circle is 2π radians, we need to multiply 3.5 by 2π to find the radian measure of the angle. continued on next slide

A Ferris wheel has a radius of 30 feet and is rotating at 3.5 revolutions per minute. Find the linear speed, in feet per minute, of a seat on the Ferris wheel. This will give us a radian angle measure of 7π. We now use that in the formula for s to find the length of the circular arc. Now the amount of time that the Ferris wheel took to go that 210π feet was 1 minute. This means that to find the linear speed, we divide the distance by the time it took to travel that distance. This will give us: