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# How to Use Tangent Table

If you are a student or professional who deals with trigonometric functions, then you must know how important it is to use a tangent table. Tangent is one of the three major trigonometric functions, along with sine and cosine. It is used to calculate the slope of a line and the angle of elevation or depression. In this article, we will discuss how to use a tangent table effectively.

## What is a Tangent Table?

A tangent table is a table that lists the values of the tangent function for different angles. It is a useful tool for calculating the tangent of an angle without using a calculator. The table contains columns for angles in degrees and their corresponding tangent values. It is similar to a logarithmic table or a multiplication table.

## How to Read a Tangent Table

To use a tangent table, you need to know the angle in degrees whose tangent value you want to find. For example, if you want to find the tangent of 30 degrees, you need to look for the row that contains 30 degrees in the first column. Once you find the row, look for the column that corresponds to the nearest tangent value.

For example, the tangent of 30 degrees is 0.577. If you look at the tangent table, you will find that the tangent of 29 degrees is 0.554 and the tangent of 31 degrees is 0.583. Since 30 degrees is closer to 29 degrees than 31 degrees, we use the tangent value of 29 degrees, which is 0.554.

## How to Use Interpolation

Sometimes, the angle you want to find may not be listed in the tangent table. In this case, you need to use interpolation to estimate the tangent value. Interpolation is a mathematical technique used to estimate values between known data points.

To use interpolation, you need to find two values in the table that bracket the angle you want to find. Then, use the formula (tan a + (tan b – tan a) / (b – a) * (x – a)), where a is the smaller angle, b is the larger angle, and x is the angle you want to find.

For example, if you want to find the tangent of 37 degrees, you can use the values of 36 degrees and 38 degrees. The tangent value of 36 degrees is 0.726 and the tangent value of 38 degrees is 0.781. Using the formula, we get:

tan 37 = tan 36 + (tan 38 – tan 36) / (38 – 36) * (37 – 36)
= 0.726 + (0.781 – 0.726) / 2 * (37 – 36)

= 0.7535

Therefore, the tangent of 37 degrees is 0.7535.

## How to Use Tangent Table for Trigonometric Equations

A tangent table is also useful for solving trigonometric equations. For example, consider the equation tan x = 1. To find the solution, we need to find the angle whose tangent value is 1.

From the tangent table, we know that the tangent of 45 degrees is 1. Therefore, the solution to the equation is x = 45 degrees or pi/4 radians.

Similarly, if we want to solve the equation tan x = -1, we need to find the angle whose tangent value is -1. From the tangent table, we know that the tangent of 135 degrees is -1. Therefore, the solution to the equation is x = 135 degrees or 3*pi/4 radians.

## How to Use Tangent Table for Trigonometric Identities

Tangent table is also useful for verifying trigonometric identities. For example, consider the identity: sin^2 x + cos^2 x = 1. We can rewrite this identity as:

sin^2 x = 1 – cos^2 x

Dividing both sides by cos^2 x, we get:

tan^2 x + 1 = sec^2 x

This is a well-known identity in trigonometry. We can use the tangent table to verify this identity for a specific angle. For example, let's verify this identity for x = 30 degrees.

From the tangent table, we know that the tangent of 30 degrees is 0.577. Therefore, the tangent squared is 0.577^2 = 0.333.

The cosine of 30 degrees is 0.866. Therefore, the secant squared is 1/cos^2 x = 1/0.866^2 = 1.314.

Substituting these values in the identity, we get:

0.333 + 1 = 1.314

This is true, which verifies the identity for x = 30 degrees.

## Conclusion

In conclusion, a tangent table is a useful tool for calculating the tangent of an angle without using a calculator. To use a tangent table, you need to know the angle in degrees whose tangent value you want to find. If the angle is not listed in the table, you can use interpolation to estimate the tangent value. A tangent table is also useful for solving trigonometric equations and verifying trigonometric identities.

## FAQs

- What is the range of values in a tangent table?

The range of values in a tangent table is from -infinity to +infinity.

- How can I create my own tangent table?

You can create your own tangent table by calculating the tangent values for different angles using a calculator or a computer program.

- Can I use a tangent table for non-right triangles?

No, a tangent table is only applicable for right triangles.

- Can I use a tangent table for negative angles?

Yes, a tangent table can be used for negative angles. The tangent of a negative angle is equal to the negative of the tangent of its corresponding positive angle.

- How can I memorize the values in a tangent table?

It is not necessary to memorize the values in a tangent table. With practice, you can learn to read and interpret the table quickly and accurately.