## TRIGONOMETRY

Trigonometry is the branch of mathematics that studies the relationships involving lengths of sides and measures of angles of triangles.

It is a useful tool for engineers, scientists, and surveyors and is applied even in seismology and navigation.

## Angles

The angle AOB is a measure formed by two rays OA and OB sharing the common point O as shown below.

The common point O is called the vertex of the angle.

If we rotate the ray OA about its vertex O and takes the position OB, then OA and OB respectively are called the initial side and the terminal side of the angle produced.

An anticlockwise rotation generates a positive angle (angle with positive sign), while a clockwise rotation generates a negative angle (angle with negative sign).

One full anticlockwise (or clockwise) rotation of OA back to itself is called one complete rotation or revolution.

## Degree Measure

The degree is a unit of measurement of angles and is represented by the symbol °.

In degrees, we split up one complete rotation into 360 equal parts and each part is one degree, denoted by 1°. Thus, 1° is 1/360 of one complete rotation.

To measure a fraction of an angle and also for accuracy of measurement of angles, minutes and seconds are introduced. One minute (1') corresponds to 1/60 of a degree and in turn a second (1'') corresponds to 1/60 of a minute (or) 1/3600 of a degree.

## Pair of Angles - Classification

We shall classify a pair of angles in the following way for better understanding and usages.

(i) Two angles that have the exact same measure are called congruent angles.

(ii) Two angles that have their measures adding to 90° are called complementary angles.

(iii) Two angles that have their measures adding to 180° are called supplementary angles.

(iv) Two angles between 0° and 360° are conjugate if their sum equals 360°.

## Angles in Standard Position

An angle is said to be in standard position, if its vertex is at the origin and its initial side is along the positive x-axis.

An angle is said to be in the first quadrant, if in the standard position, its terminal side falls in the first quadrant. Similarly, we can define for the other three quadrants.

Angles in standard position having their terminal sides along the x-axis or y-axis are called quadrantal angles.

Thus, 0°, 90°, 180°, 270° and 360° are quadrantal angles.

## Coterminal Angles

One complete rotation of a ray in the anticlockwise direction results in an angle measuring of 360°.

By continuing the anticlockwise rotation, angles larger than 360° can be produced.

If we rotate in clockwise direction, negative angles are produced.

Angles 57°, 417° and −303° have the same initial side and terminal side but with different amount of rotations, such angles are called coterminal angles.

Thus, angles in standard position that have the same terminal sides are coterminal angles . Hence, if α and β are coterminal angles, then

=  α + k(360°), k is an integer

The measurements of coterminal angles differ by an integral multiple of 360°.

For example, 417° and −303° are coterminal because

417° − (−303°)  =  720°  =  2(360°)

Note :

(i) Observe that 45°, −315° and 405° lie in the first quadrant.

(ii) The following pairs of angles are coterminal angles (30°, 390°) ; (280°, 1000°) and (−85°, 275°).

The radian measure of an angle is the ratio of the arc length it subtends, to the radius of the circle in which it is the central angle.

Consider a circle of radius r. Let s be the arc length subtending an angle θ at the centre.

Then,

Hence,

s  =  rθ

## Relationship between Degree and Radian Measures

We have degree and radian units to measure angles. One measuring unit is better than another if it can be defined in a simpler and more intuitive way.

For example, in measuring temperature, Celsius unit is better than Fahrenheit as Celsius was defined using 0° and 100° for freezing and boiling points of water.

Radian measure is better for conversion and calculations. Radian measure is more convenient for analysis whereas degree measure of an angle is more convenient to communicate the concept between people.

Greek Mathematicians observed the relation of π which arises from circumference of a circle and thus, π plays a crucial role in radian measure.

In unit circle, a full rotation corresponds to 360° whereas, a full rotation is related to 2π radians, the circumference of the unit circle.

Thus, we have the following relations :

Divide each side by 2.

Then,

Observe that the scale used in radians is much smaller than the scale in degrees. The smaller scale makes the graphs of trigonometric functions more visible and usable. The above relation gives a way to convert radians into degrees or degrees into radians.

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

If you have any feedback about our math content, please mail us :

v4formath@gmail.com

You can also visit the following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6