CBSE NCERT Homework Help: HOW TO PREPARE TRIGONOMETRY TABLE

Friday 6 September 2013

HOW TO PREPARE TRIGONOMETRY TABLE

In case your scientific calculator decides to conk out on you, you can construct a table that shows the exact values of trigonometry functions for the most commonly used angles. A bonus to this situation is that your trig table gives you exact values — the scientific calculator gives you decimal numbers that are usually rounded to some value.

In trig, you frequently use the exact values of the most favorite angles because they give better results in computations and applications, so memorizing those exact values is always a good idea.

A quick, easy way to memorize the exact trig-function values of the most common angles is to construct a table, starting with the sine function and working with a pattern of fractions and radicals. Create a table with the top row listing the angles, as shown in the following figure. The first function, in the next row, is sine.

The entries following sin in the second row are the fractions and radicals with the following pattern:

Each fraction has a denominator of 2.

The numerators of the fractions are radicals with 0, 1, 2, 3, and 4 under them, in that order, as shown in the following figure.

Next, simplify the fractions that can be simplified

so the table becomes what you see in the following figure:

The next row, for the cosine, is just the sine’s row in reverse order, as shown next.

The next row is for the tangent. In a right triangle, you find the tangent of an acute angle with the ratio

You get the same ratio when you divide sine by cosine. Here’s why it works:

Because you already know the values for sine and cosine, you can use this property (tangent equals sine divided by cosine) to get the tangent values for the table:

which is undefined. So the tangent of 90 degrees doesn’t have a value — it simply doesn’t exist.

See the following figure for the completed table with the tangent row.

QUESTION ON TRIGONOMETRY

Solve for x in the following equation.

Example 1:

There are an infinite number of solutions to this problem. To solve for x, you must first isolate the sine term.

We know that the therefore The sine function is positive in quadrants I and II. The is also equal to Therefore, two of the solutions to the problem are and

The period of the sin function is This means that the values will repeat every radians in both directions. Therefore, the exact solutions are and where n is an integer. The approximate solutions are and where n is an integer.

These solutions may or may not be the answers to the original problem. You much check them, either numerically or graphically, with the original equation.

Numerical Check:

Check answer .

Left Side:

Right Side: 0

Since the left side equals the right side when you substitute for x, then is a solution.

Check answer .

Left Side:

Right Side: 0

Since the left side equals the right side when you substitute for x, then is a solution.

Graphical Check:

Graph the equation

f (x) = 2 sin(x) - 1

Note that the graph crosses the x-axis many times indicating many solutions. Note that it crosses at . Since the period is , it crosses again at 0.5236+6.283=6.81 and at0.5236+2(6.283)=13.09, etc. The graph crosses at . Since the period is , it will cross again at 2.618+6.283=8.9011 and at 2.618+2(6.283)=15.18, etc.

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