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## Class 11 Physics (India)

### Unit 3: Lesson 4

Basics of trigonometry

# Trig ratios of special triangles

Learn to find the sine, cosine, and tangent of 45-45-90 triangles and also 30-60-90 triangles.
Until now, we have used the calculator to evaluate the sine, cosine, and tangent of an angle. However, it is possible to evaluate the trig functions for certain angles without using a calculator.
This is because there are two special triangles whose side ratios we know! These two triangles are the 45-45-90 triangle and the 30-60-90 triangle.

## The special triangles

30-60-90 triangles
A 30-60-90 triangle is a right triangle with a 30, degrees degree angle and a 60, degrees degree angle.
45-45-90 triangles
A 45-45-90 triangle is a right triangle with two 45, degrees degree angles.

## The trigonometric ratios of $30^\circ$30, degrees

We are now ready to evaluate the trig functions of these special angles. Let's start with 30, degrees.
Study the worked example below to see how this is done.

### What is $\sin(30^\circ)$sine, left parenthesis, 30, degrees, right parenthesis?

Here's a worked example:
Step 1: Draw the special triangle that includes the angle of interest.
Step 2: Label the sides of the triangle according to the ratios of that special triangle.
Step 3: Use the definition of the trigonometric ratios to find the value of the indicated expression.
\begin{aligned} \sin (30^\circ) &= \dfrac{\text{opposite }}{\text{hypotenuse}} \\\\ &= \dfrac{x}{2x} \\\\ &= \dfrac{1\maroonD{\cancel{x}}}{2\maroonD{\cancel{x}}} \\\\ &=\dfrac{1}{2}\end{aligned}
Note that you can think of x as 1, x so that it is clear that start fraction, x, divided by, 2, x, end fraction, equals, start fraction, 1, x, divided by, 2, x, end fraction, equals, start fraction, 1, divided by, 2, end fraction.
Now let's use this method to find cosine, left parenthesis, 30, degrees, right parenthesis and tangent, left parenthesis, 30, degrees, right parenthesis.

## The trigonometric ratios of $45^\circ$45, degrees

Let's try this process again with 45, degrees. Here we can start by drawing and labeling the sides of a 45-45-90 triangle.

## The trigonometric ratios of 60$^\circ$degrees

The process of deriving the trigonometric ratios for the special angles 30, degrees, 45, degrees, and 60, degrees is the same.
While we have not yet explicitly shown how to find the trigonometric ratios of 60, degrees, we have all of the information we need!

## A summary

We have calculated the trig ratios for 30, degrees, 45, degrees, and 60, degrees. The table below summarizes our results.
cosine, left parenthesis, theta, right parenthesissine, left parenthesis, theta, right parenthesistangent, left parenthesis, theta, right parenthesis
theta, equals, 30, degreesstart color #1fab54, start fraction, square root of, 3, end square root, divided by, 2, end fraction, end color #1fab54start color #1fab54, start fraction, 1, divided by, 2, end fraction, end color #1fab54start color #1fab54, start fraction, square root of, 3, end square root, divided by, 3, end fraction, equals, start fraction, 1, divided by, square root of, 3, end square root, end fraction, end color #1fab54
theta, equals, 45, degreesstart color #aa87ff, start fraction, square root of, 2, end square root, divided by, 2, end fraction, equals, start fraction, 1, divided by, square root of, 2, end square root, end fraction, end color #aa87ffstart color #aa87ff, start fraction, square root of, 2, end square root, divided by, 2, end fraction, equals, start fraction, 1, divided by, square root of, 2, end square root, end fraction, end color #aa87ffstart color #aa87ff, 1, end color #aa87ff
theta, equals, 60, degreesstart color #1fab54, start fraction, 1, divided by, 2, end fraction, end color #1fab54start color #1fab54, start fraction, square root of, 3, end square root, divided by, 2, end fraction, end color #1fab54start color #1fab54, square root of, 3, end square root, end color #1fab54
These values tend to occur often in advanced trigonometry problems. Because of this, it is helpful to know them.
Some people choose to memorize these values, but memorization is not necessary. In this article, you derived the values yourself, so hopefully you can re-derive them whenever you need them in the future.