Trigonometric Ratios

Trigonometric Ratios are defined as the values of all the trigonometric function based on the value of the ratio of side in a right-angled triangle.

Definition 1: The ratios of sides of a right-angled triangle with respect to any of its acute angles are known as the trigonometric ratios of that particular angle.

Consider a right-angled triangle, right-angled at B.

Right-Angled Triangle

With respect to \(\angle C\), the trigonometric ratios are given as:

  • sine: sine of an angle is defined as the ratio of the side opposite(perpendicular side) to that angle to the hypotenuse.
  • cosine: cosine of an angle is defined as the ratio of the side adjacent to that angle to the hypotenuse.
  • tangent: tangent of an angle is defined as the ratio of the side opposite to that angle to the side adjacent to that angle.
  • cosecant: cosecant is a multiplicative inverse of sine.
  • secant: secant is a multiplicative inverse of cosine.
  • cotangent: cotangent is multiplicative inverse of tangent.

The above ratios are abbreviated as sin, cos, tan, cosec, sec and tan respectively in the order they are described. So, for Δ ABC, trigonometric ratios are defined as:

\( \sin \angle C \) = \( \frac {Side~opposite~to~\angle C}{Hypotenuse}\) = \( \frac {AB}{AC} \)

\( cos ~\angle C \) = \( \frac {Side~adjacent~to~\angle C}{Hypotenuse}\) = \( \frac {BC}{AC} \)

\( tan ~\angle C \) = \( \frac {Side~opposite~to~\angle C}{Side~adjacent~to~\angle C}\) = \( \frac {AB}{AC} \) = \( \frac {sin~\angle C}{cos~\angle C}\)

\( cosec ~\angle~C\) = \( \frac {1}{sin~\angle~C} \) = \( \frac {Hypotenuse}{Side~opposite~to~\angle~C}\)=\( \frac {AC}{AB}\)

\( sec ~\angle~C\) = \( \frac {1}{cos~\angle~C} \) = \( \frac {Hypotenuse}{Side~adjacent~to~\angle~C}\)=\( \frac {AC}{BC}\)

\( cot ~\angle~C\) = \( \frac {1}{tan~\angle~C} \) = \( \frac {Side~adjacent~to~\angle~C}{Side~opposite~to~\angle~C}\)= \( \frac {BC}{AB}\)

In right Δ ABC , if ∠and ∠C are assumed as \( 30^\circ \)  and \( 60^\circ \), then there can be infinite right triangles with those specifications but all the ratios written above for ∠C in all of those triangles will be same. So, all the ratios for any of the acute angles (either ∠A or ∠C) will be same for every right triangle. This means that the ratios are independent of lengths of sides of the triangle.

Application of Trigonometry-

Trigonometry is one of the most important branches of mathematics. Some of the applications of Trigonometry are:

  • Measuring the heights of towers or big mountains
  • Determining the distance of the shore from sea
  • Finding the distance between two celestial bodies
  • Determining the power output of solar cell panels at different inclinations
  • Representing different physical quantities such as mechanical waves, electromagnetic waves, etc.

It is evident from the above examples that trigonometry has its involvement in a major part of our day-to-day life and much more. In most of the applications listed above, something was being measured and that is what trigonometry is all about. Actually, it is composed of two Greek words: Trigon and Metron, trigon meaning ‘triangle’ and metron meaning ‘to measure’. It is the study of sides and angles of triangles and relationships between them.

 

Since the values of trigonometric ratios are constant for a given angle, they can be determined for some specific angles using properties of triangles. To learn more about trigonometry formulas, trigonometry table and thousands of concepts in an easy and fun way, visit BYJU’S.


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tan1.tan89=tan2.tan88