TUC-1 Measurements of Angles “Things I’ve Got to Remember from the Last Two Years”

TUC-1 Measurements of Angles

“Things I’ve Got to Remember from the Last Two Years”

PrecalculusMPH 9/11

The Coordinate Plane

In the coordinate plane, angles in standard position are created by rotating about the origin (the vertex of the angle). The initial ray is the positive x –axis.

Initial Ray

Terminal Ray

Positive Rotation – counterclockwise

Negative Rotation - clockwise

PrecalculusMPH 9/11

The Radian

Angles can also be measured in radians.

A central angle measures one radian when the measure of the intercepted arc equals the radius of the circle.

In the circle shown, the length of the intercepted arc equals the radius of the circle. Hence, the angle theta measures 1 radian.

r

r

r

Wait! What is a radian?

Visual animation of what a radian represents.

This visual was created by

LucasVB, this is a link to his blog post about radians

CRO 8/14 Precalculus

PrecalculusMPH 9/11

Radians

If one investigated one revolution of a circle, the arc length would equal the circumference of the circle. The measure of the central angle would be 2 radians.

Since 1 revolution of a circle equals 360,2 radians = 360!!

PrecalculusMPH 9/11

Radians

This implies that 1 radian 57.2958.

The coordinate plane now has the following labels.

0, 0

90, /2

180, 360, 2

270, 3/2

PrecalculusMPH 9/11

Converting from Degrees to Radians

To convert from degrees to radians, multiply by

Example 1 Convert 320 to radians.

Example 2 Convert -153 to radians.

PrecalculusMPH 9/11

Converting from Radians to Degrees

To convert from degrees to radians, multiply by

Example 1 Convert to degrees.

PrecalculusMPH 9/11

Converting from Radians to Degrees

Example 2 Convert to degrees.

Example 3 Convert 1.256 radians to degrees.

PrecalculusMPH 9/11

Coterminal Angles

Angles that have the same initial and terminal ray are called coterminal angles.

Graph 30 and 390 to observe this.

Coterminal angles may be found by adding or subtracting increments of

360 or 2

PrecalculusMPH 9/11

Coterminal Angles

Example 1

Find two coterminal angles (one positive and one negative) for 425.

425 - 360 = 65 65 - 360 = -295

The general expression would be:

425 + 360n where n integer (I)

PrecalculusMPH 9/11

Coterminal Angles

Example 2

Find two coterminal angles (one positive and one negative) for

The general expression would be:

PrecalculusMPH 9/11

Coterminal Angles

Example 3

Find two coterminal angles (one positive and one negative) for -3.187R.

-3.187 – 2π = -9.470R

-3.187 + 2π = 3.096R

The general expression would be:

-3.187 + 2πn where n I

PrecalculusMPH 9/11

Complementary Angles

Two angles whose measures sum to 90 or /2 are called complementary angles.

The complement of 37 is 53.

The complement of /8 is 3/8.

The complement of 1.274R is 0.297R.

PrecalculusMPH 9/11

Supplementary Angles

Two angles whose measures sum to 180 or are called supplementary angles.

The supplement of 85 is 95.

The supplement of 217 does not exist. Why?

PrecalculusMPH 9/11

Supplementary Angles

The supplement of /8 is 7/8.

The supplement of 2.891R is 0.251R.