TRIGONOMETRY INTRODUCTION

 In this post, we'll learn about Trigonometric ratios of acute angles.

An angle is said to be acute if it is greater than 0° but less than 90°. Consider a right-angled triangle ABC with right angle at B.

We know that,

The side opposite to the right angle is the hypotenuse. The side opposite to angle A is called as opposite. The side adjacent to angle A is called as adjacent.

Consider the triangle ABC as shown in the figure,


The ratio of lengths of any two sides of the triangle can be determined by the angle A and are independent of size of the triangle. The number of such possible ratios are 6. Each of them has a name as follows,

\textup{sin}\, A=\frac{\textup{opposite}}{\textup{hypotenuse}}=\frac{\textup{o}}{\textup{h}}

\textup{cos}\, A=\frac{\textup{adjacent}}{\textup{hypotenuse}}=\frac{\textup{a}}{\textup{h}}

\textup{tan}\, A=\frac{\textup{opposite}}{\textup{adjacent}}=\frac{\textup{o}}{\textup{a}}

\textup{cot}\, A=\frac{\textup{adjacent}}{\textup{opposite}}=\frac{\textup{a}}{\textup{o}}

\textup{sec}\, A=\frac{\textup{hypotenuse}}{\textup{adjacent}}=\frac{\textup{h}}{\textup{a}}

\textup{cosec}\, A=\frac{\textup{hypotenuse}}{\textup{opposite}}=\frac{\textup{h}}{\textup{o}}


The ratios sin, cos, tan, cot, sec and cosec stand for sine, cosine, tangent, cotangent, secant, cosecant of angle A respectively. These functions of angle A are called trigonometric ratios.

The trigonometric ratios of some standard angles are given below.





We know that, an identity is an equation which holds true for every value of occurring variables. Therefore, a trigonometric identity is a trigonometric equation which holds true for every value of occurring value.

Some of the trigonometric identities are given below,


\\sin^2\theta+cos^2\theta=1\\\\sec^2\theta-tan^2\theta=1\\\\cosec^2\theta-cot^2\theta=1


The above equations hold true irrespective of value of the angle.




The formulas for Trigonometric ratios of compound angles are as follows,


Cosine of the sum and difference of two angles

\\cos(A+B)=cos\, A \, cos\, B-sin\, A\, sin\, B \\\\cos(A-B)=cos\, A \, cos\, B+sin\, A\, sin\, B


Sine of the sum and difference of two angles

\\sin(A+B)=sin\, A \, cos\, B+cos\, A\, sin\, B \\\\sin(A-B)=sin\, A \, cos\, B-cos\, A\, sin\, B


Tangent of the sum and difference of two angles

\\\\tan(A+B)=\frac{tanA+tanB}{1-tanA\, tanB} \\\\\\tan(A-B)=\frac{tanA-tanB}{1+tanA\, tanB}


Some more important results

\\1. \; sin(A+B)\, sin(A-B)=sin^2A-sin^2B=cos^2B-cos^2A \\\\\\2. \; cos(A+B)\, cos(A-B)=cos^2A-sin^2B=cos^2B-sin^2A \\\\\\3.\; sin(A+B+C)=\left (sinA\, cosB\, cosC+cosA\, sinB\, cosC+cosA\, cosB\, sinC-sinA\, sinB\, sinC \right ) \\\\\\4.\; cos(A+B+C)=\left (cosA\, cosB\, cosC-cosA\, sinB\, sinC-sinA\, cosB\, sinC-sinA\, sinB\, cosC \right ) \\\\\\5.\; tan(A+B+C)=\frac{tanA+tanB+tanC-tanA\, tanB\, tanC}{1-tanA\, tanB-tanB\, tanC-tanC\, tanA}




Formulas to transform the product into sum or difference


\\2sinA\, cosB=sin(A+B)+sin(A-B) \\\\\\2cosA\, sinB=sin(A+B)-sin(A-B) \\\\\\2cosA\, cosB=cos(A+B)+cos(A-B) \\\\\\2sinA\, sinB=cos(A-B)-cos(A+B)


Formulas to transform the sum or difference into product


\\sinC+sinD=2sin\left ( \frac{C+D}{2} \right )cos\left ( \frac{C-D}{2} \right ) \\\\\\\\sinC-sinD=2cos\left ( \frac{C+D}{2} \right )sin\left ( \frac{C-D}{2} \right ) \\\\\\\\cosC+cosD=2cos\left ( \frac{C+D}{2} \right )cos\left ( \frac{C-D}{2} \right ) \\\\\\\\cosC-cosD=2sin\left ( \frac{C+D}{2} \right )sin\left ( \frac{D-C}{2} \right )

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