Measures of Angles

There are three standard systems of measuring angles, namely, the revolution system, the degree system and the radian system.

A simple but logical system is the revolution system. In this system, an angle is measured by the number of revolution or fraction of a revolution made by the terminal side in relation to the initial side. The unit of measurement in this system is revolution (rev) which is equivalent to one whole angle.

The second system of measuring angle is the degree system. It is patterned after the sexagesimal system of ancient Babylon. In this system the units of measurement is in degree (°). A degree is an angle formed by a rotation about its vertex equal to 1/360 of a complete revolution. The whole angle is divided into 360 degrees or 360° a number that is multiple of 60.

The degree is further divided into minutes ( ' ) and minutes into seconds ( " ) with the following conversions:

1° = 60' (min.)
1' (min.)= 60" (sec.)

1° = 3600" (sec.)

The third system of measuring angle is based on the relation between the radius an the circumference of a circle. This is known as the radian system wherein the unit of measurement is the radian (rad). An angle of one radian is a central angle describe by an arc whose length is equal to the radius of the circle.








CONVERSION TABLE:

RADIAN TO REVOLUTION


1rad = 1/2π rev.

2π = 1rev.

RADIAN TO DEGREE


π rad = 180°


DEGREE TO REVOLUTION


1° = 1/360 revolution

INTRODUCTION TO PLANE TRIGONOMETRY

Definition:

Trigonometry come from two Greek words: trigonon, which means "triangle", and metron, meaning "measure". Thus trigonometry is concerned with the measurement of angles and the side of the triangle.

Angles
In geometry we learned that angles are geometric figures formed when two lines(rays) meet at one common point



An angle is the figure formed or generated by the rotation of a line segment around a fixed point.

The first position is called initial side, while the final position is called terminal side.
the fixed point is the vertex of the angle. The arrowhead indicate the direction of the rotation.

The angle above is rotating counter clockwise there fore it is a positive angle.

If the angle is rotating clockwise it is a negative angle.



TYPES OF ANGLE

1. Zero Angle - angle whose measure is 0°
2. Acute Angle - angle whose measure is 0°< θ <90°
3. Right Angle - angle whose measure is exactly 90°
4. Obtuse Angle - angle whose measure is 90° < θ <180°
5. Straight Angle - angle whose measure is exactly 180°
6. Reflex Angle - angle whose measure is 180° < θ <360°
7. Complex Angle - angle whose measure is greater than 360°
8. Revolution Angle- angle whose measure is 360°

An angle in standard position is an angle having its initial side lying on the positive x- axis.



Coterminal angles are angles having the same terminal sides.

Example:
given: θ = 45°



Its co-terminal angles are 405°, 765°, -315° since they have the same terminal sides.

HINT: If your ask to find the co-terminal angles of a given angle, simply add or subtract the given angle by 360°.

for example you're ask to find the lowest positive and highest negative co-terminal angles of 30°.

simply add the given angle which is 30° by 360°

30° + 360° = 390° so the lowest positive co-terminal angle of 30° is 390°

for the highest negative co-terminal.
simply subtract the given angle which is 30° by 360°

30° - 360° = -330° so the highest negative co-terminal angle of 30° is -330°.


Quadrantal angle is an angle in standard postion having its terminal side lying on either x or y axis. like 90°, 180°, 270°, 360°,