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Trigonometry Rocks my World

Basic trigonometry is based on similar triangles. For example, lets say there are two similar triangles with the lengths 3,4,5, and another with the lengths 6,8,10. The ratios of the lengths of the twotriangles are 1:2. The result is that corresponding angles between the two triangles will be the same. This concept is called trigonometric ratios.

The History of Trigonometry

The hypotenuse is the side opposite the rightangle. In all basic trigonometric problems one of the other angles wil be important. The side opposite this angle isknown as the opposite side. The third side nect tothis angle is knownas the adjacent side. The main trigonometric functions are defined as

sin@=opp/hyp

cos@=adj/hyp

tan@=opp/adj

Sine, Cosine, and Tangent Formulas

In the case of the sine and cosine functions, in y=a sin x, the amplitude becomes a and not 1. This dilation will not affect the shape of the graph. For example, for y=2cosx, the height of the graph will be twice as long as 1cosx

y=-cosx is the cosine graph reflected in the x-axis. This is because the y values will be reversed in sign from thos calculated for the cosine graph.

However, for sine, y= sin(-x) will reflect the basic sine graph in the y-axis. Functions of the form y=cos(nx) are dilated parallel to the x-axis which means the period of the graph will be altered. A general function od y=a sin (bx) + c has: an amplitude of "a", a period of 360/b, ad a translation of "c" units upwards


Sin, Cos and Tan Functions


The trigonometric functions of angles are the ratios of the various sides of a triangle. Consider a right-angled triangle ABC as shown in the figure below.


The following terminology is useful.
  • Hypotenuse: The side opposite to the right angle in a triangle is called the hypotenuse. Here the side AC is the hypotenuse.

  • Opposite Side: The side opposite to the angle in consideration is called the opposite side. So, if we are considering angle A, then the opposite side is CB.

  • Base: The third side of the triangle, which is one of the arms of the angle under consideration, is called the base. If A is the angle under consideration, then the side AB is the base.


For angle A (sometimes referred to as angle CAB), the following fundamental trigonometric functions can be defined.

Sine of A = sin A = Opposite Side / Hypotenuse = CB/CA = a/b
Cosine of A = cos A = Base / Hypotenuse = AB/CA = c/b
Tangent of A = tan A = Opposite Side / Base = CB/AB = a/c

From the definitions, it can be seen that tan A = sin A / cos A.


Cosec, Sec and Cot Functions


A few more useful functions can be defined.

Cosecant of A = cosec A = 1 / sin A = b/a
Secant of A = sec A = 1 / cos A = b/c
Cotangent of A = cot A = 1 / tan A = c/a

From the definitions, it can be seen that cot A = cosec A / sec A


 
 

Sine and Cosine Rules

The three functions above are the basis for solving right angles triangles and have a number of applications including navigation and surveying history Ptolemy was the author of the book of chord which divided the circle into 360 degrees and the diameter into 120 parts. Originally, the term chords was used to represent the idea of sine and cosine angles by Ptolemy as well as Hipparchus, both of whom were famous greek mathematicians.

Trigonometry is based on the measurement of angles. A circle is divided into 360 degrees so that a right angle is 90 degrees.

The concept of Angles

Trionometric ratios of all angles are done with th use of a unit circle. Negative angles are made going clockwise from the positice direction of the X-axis. There is an infinate set of angles which give valuse for the main trigonometric ratios. They are all separated by one complete turn on the unit circle.

Angles are almost always measured with radians

Interesting Trig Image

Trig Fundementals