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How to Create a Table of Trigonometry Functions

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.


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