Trigonometry (Introduction)

Welcome to *the only place where the essence of trigonometry is explained*.

• a right-triangle is specified by 2 independent parameters.

• That means, if an angle and the length of a side is given, then one should be able to calculate the length of the other two sides.

• What property one can use to calculate the above? For a given angle, the ratio of sides is a constant.

Thus, the ratio of sides comes into existence as $\mathrm{sin}$, $\mathrm{cos}$, $\mathrm{tan}$ etc.

Beyond the definitions of trigonometric ratios, the following are covered.

• trigonometric ratios for standard angles

• trigonometric identities based on Pythagoras Theorem

Basics of Angles

In this page, the basics of Angles required to understand trigonometry is revised.

Basics of Triangles

In this page, the basics of triangles required to understand trigonometry is revised.

Importance of Right Angled Triangle

In this page, the importance of right angled triangles in application is explained. This justifies that problems of any polygon is simplified into right-triangles.

Trigonometric Ratios Explained

This is the only place in the world to provide fundamental explanation to trigonometric ratios.

» An angle $\theta$ specifies a class of similar-right-triangles

» a side narrows down to a specific right-triangle

» Given an angle and a side "How to compute other sides?"

» Ratio of sides of similar triangles for an angle $\theta$

→ $\frac{\text{side1}}{\text{side3}}=\text{constant1}$

→ $\frac{\text{side2}}{\text{side3}}=\text{constant2}$

→ $\frac{\text{side1}}{\text{side2}}=\text{constant3}$

» Any parameter of right-triangles (sides and angles) can be calculated using the ratios : **Trigonometric Ratios**

Triangular Form of Trigonometric Ratios

In this page, the trigonometric ratios defined in triangular form is revised.

Note: The trigonometric ratios are also called trigonometric values and are defined in unit circle form. This will be explained in advanced trigonometry.

Understanding Standard Angles

What are the standard angles for which trigonometric ratios are defined? These angles are chosen because of some pattern or properties. This page explains the reason why some angles are special.

Trigonometric Ratios for Standard Angles

Students need not memorize a table of trigonometric ratios for standard angles and instead they can quickly calculate the ratios for standard angle. This page explains how to quickly calculate.

Pythagorean Trigonometric Identities

The relationship between trigonometric ratios per Pythagorean theorem is explained and referred as "Pythagorean Trigonometric Identities"

Trigonometric Ratios for Complementary Angles

The relationship between trigonometric ratios of an angle and its complementary angle is explained.