# RADIAN MEASURE AND ITS APPLICATIONS

A **radian** is an angle, which, if its vertex is

placed at the center of a circle , intercepts an

arc equal in length to the radius of the circle.

Note that

2π radians = 1 revolution = 360°and so π radians = 180°.

Thus, 1 radian = 180°/π ≈ 57°.

**Very Important:** When no unit of measure is indicated on an angle, the

angle is assumed to be expressed in radians.^{#}

30° means the measure of an angle is 30 degrees;

30 means the measure of an angle is 30 radians.

**EXAMPLE 1.** Convert 7π/9 to degrees.

**EXAMPLE 2.** Convert 6.75° to radians.

# Radian measure is a pure measurement (no units). The word "radian" can be
dropped from and

inserted into calculations without affecting the value of the expression

**EXAMPLE 3.** Find the exact value of sin 5π/6

**EXAMPLE 4**. Find the exact value of cos π.

**EXAMPLE 5.** Approximate sin 3 to four places after
the decimal point.

Arc length formula : In a circle of radius r, let an

arc of length s be intercepted by a central

angle θ, measured in radians. Then

s = rθ.

**Note:** r and s must be measured in the same units.

**EXAMPLE 6.** The radius of a circle is 100
centimeters. If A and B are two

points on the circle , and if the tangent lines at these points intersect

at point T such that angle ATB is 36°, find the length of minor arc AB.

**EXAMPLE 7.** Assuming that the earth is a sphere of
radius 4000 miles,

find the latitude of Edinburg if Edinburg is located 1800 miles above

the equator. Round to the nearest degree.

**Angular and linear velocity :** Consider a point P
moving with constant

speed on the circumference of a circle with radius r and center O. If

P traverses a distance of s linear units in t time units, then s/t =v is

called the ** linear velocity ** of P. If the radius OP swings through θ

angular units in t time units, then θ/t = ω is called the a**ngular
velocity** of P. Moreover, if θ is measured in radians and ω is in

radians per time unit, then dividing both sides of s = rθ by t, we get

v = rω.

**EXAMPLE 8.** The wheels of a bus have a diameter of
80 centimeters and

are rotating at 500 revolutions per minute. How fast is the bus

moving in kilometers per hour? Round to the nearest tenth.

**EXAMPLE 9.** A belt moving 32 feet per second is
driving a pulley at the

rate of 360 revolutions per minute. Find the radius of the pulley in

inches. Round to the nearest tenth of an inch.

**HOMEWORK**

1. Convert 6π/5 to degrees.

2. Convert 300° to radians

3. Find the exact value of tan 5π/6

4. Find the exact value of cos 3π/2

5. Find the exact value of sin 5π/4

6. In the 1840's there was a border dispute between the
United States

and Great Britain over the southern boundary of British Columbia.

The British claimed that the boundary was the 49° latitude, while the

United States said 54.7°. Assuming that the earth is a sphere of

radius 3960 miles, what was the disputed north-south distance?

Round to the nearest mile.

7. An equilateral triangle is inscribed in a circle . Find
the length of a

side of the triangle if each side subtends an arc of 28 inches. Round

to the nearest tenth of an inch.

8. The wheels of a car are rotating at 840 revolutions per
minute. If the

car is traveling at 55 miles per hour, find the diameter of the wheels in

inches. (1 mile = 5280 feet) Round to the nearest inch.

9. The pendulum of a clock swings through an arc of
precisely 8° each

second. If the tip of the pendulum travels 831 centimeters in 1

minute, how long is the pendulum? Round to the nearest tenth.

10. A truck is traveling at 120 kilometers per hour. How
many revolutions

per minute are made by the wheels, which are 80 centimeters in

diameter? Round to the nearest integer.

**ANSWERS**

6. 394 miles

7. 23.2 inches

8. 22 inches

9. 99.2 centimeters

10. 796 revolutions per minute

Prev | Next |