Understanding Trigonometry Visually

 Introduction -

Trigonometry is a branch of mathematics which studies about the relationship between the angles and side lengths of the triangles. The “Trigono” word means triangle and the “metry” word means to measure. It has various applications in navigation, astronomy, optics and so on.

 

They say Trigonometry is all about triangles, but in reality it’s about circles.

 

The following will help you visualize the different Trigonometric functions like Sine, Cosine, Tangent, Secant, Cosecant and Cotangent.

 

Visualizing Sine & cosine –

In a right angled triangle: Sine of an acute angle is defined as the ratio of length of the opposite side to its hypotenuse. And Cosine of an acute angle is defined as the ratio of length of the adjacent side to its hypotenuse. If the length of the hypotenuse is 1 unit, then the Unit circle formed by the hypotenuse will have -

`\color{orange} {\sin\theta = \frac{opposite}{hypoten use}}` 

`\color{blue} {\cos\theta = \frac{adjacent}{hypoten use}}`

Visualizing Tangent & Cotangent -

In a right angled triangle: Tangent of an acute angle is defined as the ratio of  length of the opposite side to the length of the adjacent side.

And cotangent of an acute angle is defined as the ratio of length of the adjacent side to the length of the opposite side.

`\color{red} {\tan\theta = \frac{opposite}{adjacent}}` 

`\color{black} {\cot\theta = \frac{adjacent}{opposite}}`

Visualizing Secant & Cosecant -

In a right angled triangle: secant of an acute angle is defined as the ratio of the length of the hypotenuse to the length of the adjacent side. And cosecant of an acute angle is defined as the ratio of the length of hypotenuse to the length of the opposite side. In a unit circle secant and cosecant are represented as follows -

`\color{blue} {\sec\theta = \frac{hypoten use}{adjacent}}`

`\color{orange} {cosec\theta = \frac{hypoten use}{opposite}}`