Ch 3 Radian Measure and the Unit Circle

3.1 Radian Measure

An angle with its vertex at the center of a circle that intercepts an arc on the circle equal in length to the radius of the circle has a measure of 1 radian.

radians 1 radian

Converting Between Degrees and Radians

Multiply a degree measure byradian and simplify to convert to radians.

Multiply a radian measure byand simplify to convert to degrees.

Convert 45° to radians

Convert –270° to radians

Convert 249.8° to radians

Convert radians to degrees

Convert 4.25 radians to degrees

Agreement on Angle Measurement Units

If no unit of angle measure is specified, then the angle is understood to be measured in radians.

Find tan

Find sin

Find cos

3.2 Applications of Radian Measure

The length s of the arc intercepted on a circle of radius r by a central angle of measure θ radians is given by the product of the radius and the radian measure of the angle.

(θin radians)

A circle has a radius of 18.20 cm.

Find the arc length intercepted by radians

Find the arc length intercepted by 144°

The north latitude of Reno NV is approximately 40°N while the north latitude of Los Angeles is approximately 34°N. Reno is approximately due north of L.A. Given the radius of the earth is approx 6400km, find the straightline distance.

A rope is wound around a drum with radius 0.8725 ft. How much rope is wound when rotated through 39.72°?

Two gears are adjusted so the smaller gear drives the larger one. If the smaller gear rotates through 225°, how many degrees will the larger gear rotate?

Asector of a circle is the portion of the interior of a circle intercepted by a central angle.

The area of a circle is found by .

θ = part of a circle(2π radians) of a circle

The area A of a sector of a circle of radius r and central angle θ is given by the following formula.

A center pivot irrigation system provides water to a sector shaped field rotating through 15°. Find the area supplied.

3.3 The Unit Circle and Circular Functions

A unit circle has its center at the origin and a radius of 1 unit.

The trigonometric functions of angle θ in radians found by choosing a point (x, y) on the unit circle can be rewritten as functions of the arc length s, a real number.

For any real number s represented by a directed arc on the unit circle,

Since , , and

since ,

domains:

sin and cos -

tan and sec -

cot and csc -

Find exact value of

Find exact value of and

Find exact value of

Find exact value of (120°)

calculator exercise

cos 1.85cos 0.5149 cot 1.3209

Approximate the value of when cos s = 0.9685

Find the exact value of when

tan s= 1

The angle of elevation θ of the sun in the sky at any latitude L can be found by

In Sacramento, CA, L is 38.5° (0.6720 rad). Find the angle at 3pm 29 Feb 2012, , ω=0.7854.

Suppose angle TVU measures 60°. Find lengths of OQ, PQ, VR, OV, OU, and US.

3.4Linear and Angular Speed

Linear Speed

Angular Speed

The human wrist can rotate through 90° in 0.045 seconds. Angular speed is:

Suppose P is on a circle with radius 10cm, and ray OP is rotating with angular speed radian/sec.

Find angle generated by P in 6 sec.

Find distance traveled in 6 sec.

Find linear speed of P in cm/sec.

A belt runs a pulley of radius 6cm at 80 rpm.

Find the angular speed in radians/sec.

Find the linear speed of the belt in cm/sec.

A satellite traveling in a circular orbit 1600 km above the earth takes 2 hr to make an orbit. Radius of earth is about 6400 km.

Approximate the linear speed in km/hr.

Approximate the dist traveled in 4.5 hr.