Tables of Trigonometric Functions

Let us study the Tables of Trigonometric Functions,
Calculators and tables are used to determine values of trigonometric functions. Most scientific calculators have function buttons to find the sine, cosine, and tangent of angles. The size of the angle is entered in degree or radian measure, depending on the setting of the calculator. Degree measure will be used here unless specifically stated otherwise. When solving problems using trigonometric functions, either the angle is known and the value of the trigonometric function must be found, or the value of the trigonometric function is known and the angle must be found. These two processes are inverses of each other. Inverse notations are used to express the angle in terms of the value of the trigonometric function. The expression sin θ = 0.4295 can be written as θ = Sin−10.4295 or θ = Arcsin0.4295 and these two equations are both read as “theta equals Arcsin 0.4295.” Sometimes the expression “inverse sine of 0.4295” is used. Some calculators have a button marked “arc,” which is pressed prior to the function key to express “arc” functions. Arc functions are used to find the measure of the angle if the value of the trigonometric function is known. If tables are used instead of a calculator, the same table is used for either process. Note: The use of calculators or tables gives only approximate answers.Even so, an equal (=) sign is sometimes used instead of an approximate (≈ or ≅) sign.

Example 1: What is the sine of 48°? Example 2: What angle has a cosine of 0.3912?Although a calculator can find trigonometric functions of fractional angle measure with ease, this may not be true if you must use a table to look up the values. Tables cannot list all angles. Therefore, approximation must be used to find values between those listed in the table. This method is known as linear interpolation. The assumption is made that differences in function values are directly proportional to the differences of the measures of the angles over small intervals. This is not really true, but yields a better answer than just using the closest value in the table. This method is illustrated in the following examples.

Hope the above explanation helped you.

Explain Midpoint

Let me explain you about Midpoint,

A midpoint of a line segment is the halfway point, or the point equidistant from the endpoints (Figure 1 ).

Figure 1 Midpoint of a line segment.

R is the midpoint of QS because QR = RS or because QR = ½ QS or RS = ½ QS

Example 3: In Figure 2 , find the midpoint of KR .

Figure 2 Midpoint of a line segment.

The midpoint of KR would be ½(24), or 12 spaces from either K or R. Because the coordinate of K is 5, and it is smaller than the coordinate of R (which is 29), to get the coordinate of the midpoint you could either add 12 to 5 or subtract 12 from 29. In either case, you determine that the coordinate of the midpoint is 17. That means that point O is the midpoint of KR because KO = OR.

Another way to get the coordinate of the midpoint would be to find the average of the endpoint coordinates. To find the average of two numbers, you find their sum and divide by two. (5 + 29) ÷ 2 = 17. The coordinate of the midpoint is 17, so the midpoint is point O.

Theorem : A line segment has exactly one midpoint.

Hope the above explanation helped you.

Algebraic Structure of Complex Numbers

Let us learn Algebraic structure of complex numbers,

When you come to quadratic equations you will be confronted with an entity (or non-entity) whose name is written this way - √-1, and pronounced "square root of minus one."

Complex numbers are points in the plane endowed with additional structure. We consider the set R2 = {(x, y): x, yR}, i.e., the set of ordered pairs of real numbers. Two such pairs are equal if their corresponding components coincide:

(x1, y1) = (x2, y2) iff x1 = x2 and y1 = y2.

With two operations - addition and multiplication - defined below, the set R2 becomes the set C of complex numbers. R2 considered as a set of complex numbers C is called Argand diagram or Argand plane or Gauss plane.

Hope the above explanation helped you.

Sine Waves

Let me explain what is meant by waves,

A wave is an oscillation which moves in space, such as sound waves, earthquake waves and the matter waves and light waves which turn out to make up just about everything in the universe. Sine waves turn up all over the place; more complex waveforms can always be broken down into a series of superimposed sine waves of various frequencies, in a process known as a Fourier transform. Sub-atomic 'particles' are best thought of as wave packets.

This extremely general applicability of the idea of sine waves results in trigonometric functions turning up everywhere you look in physics. The most general form of the basic wave equation, appearing everywhere from classical mechanics through electromagnetism to quantum physics, is this:

x = A × cos (ω × t + d / λ)

where λ is the wavelength (the distance between one peak of the wave and the next) and d is the distance along the wave. A full exposition of the mathematics of waves is beyond the scope of this writeup; I will just mention quickly that a fuller understanding of it requires a grasp of the idea of superposition and interference - what happens when waves meet each other; refraction - what happens when a wave passes from one medium to another; and diffraction - what happens when a wave passes through a hole. Standing waves and resonance are also profoundly important almost everywhere that waves turn up; they account for the sounds made by different objects, the energies of photon emitted by different atoms and molecules, and for a staggeringly wide range of other phenomena.

Hope the above explanation helped you..

Construction of Tangents to a Circle

Let us study about construction of tangents to a circle,
If a point lies on the
circle, then there is only one tangent to the circle at this point and it is perpendicular to
the radius through this point. Therefore, if you want to draw a tangent at a point of a
circle, simply draw the radius through this point and draw a line perpendicular to this
radius through this point and this will be the required tangent at the point.
if the point lies outside the circle, there will be two
tangents to the circle from this point.

We shall now see how to draw these tangents.

Construction : To construct the tangents to a circle from a point outside it.
We are given a circle with centre O and a point P outside it. We have to construct
the two tangents from P to the circle.
Steps of Construction:
1. Join PO and bisect it. Let M be the midpoint
of PO.
2. Taking M as centre and MO as radius, draw
a circle. Let it intersect the given circle at
the points Q and R.
3. Join PQ and PR.
Then PQ and PR are the required two
tangents (see Fig.).
Now let us see how this construction works.
Join OQ. Then ∠ PQO is an angle in the
semicircle and, therefore,
∠ PQO = 90°
Can we say that PQ ⊥ OQ?
Since, OQ is a radius of the given circle, PQ has to be a tangent to the circle. Similarly,
PR is also a tangent to the circle.
Note : If centre of the circle is not given, you may locate its centre first by taking any
two non-parallel chords and then finding the point of intersection of their perpendicular
bisectors. Then you could proceed as above.
Hope the above explanation helped you, now let me explain types of tangent lines.