Trigonometry

Trigonometry is the study of relationships between the sides and angles of a triangle. We will study some ratios of the sides of a right triangle with respect to its acute angles, called trigonometric ratios of the angle. We will restrict our learning to acute angles only.

Trigonometric Ratios:

In above right angled triangle, angle A is acute, angle C is acute and angle B = 90 degree.

Side AC is the hypotenuse of the right angled triangle and the side AB is a part of angle A. We call it the side adjacent to angle A.

Sin A ={BC}{AC}

Cos A ={AB}{AC}

Tan A ={BC}{AB}

Cosec A ={1}{Sin A}{AC}{BC}

Sec A ={1}{Cos A}{AC}{BC}

Cot A ={1}{Tan A}{Cos A}{Sin A}{AB}{BC}

Also Tan A ={BC}{AB}= {Sin A}{Cos A}

According to Pythagoras theorem in above right angled triangle ,

(AC)²  = (AB)²+ (BC)²

Dividing by  (AC)² in both sides, we get,

1 = Sin²A  + Cos²A

Since Sin A ={BC}{AC}

and Cos A = {AB}{AC}

So  Sin²A  = 1 – Cos²A, and  Cos²A = 1 – Sin²A

Similarly Sec²A  – Tan²A = 1

So Sec²A = 1 + Tan²A and Tan²A = Sec²A – 1

Also Cosec²A  – Cot²A = 1

So  Cosec²A  = 1 +  Cot²A  and Cot²A = Cosec²A  – 1

Sin(A+B) = SinA. CosB + CosA. SinB

Sin(A-B) = SinA. CosB – CosA. SinB

Cos(A+B) = CosA. CosB – SinA.SinB

Cos(A-B) = CosA. CosB + SinA.SinB

Tan(A+B) = {TanA + TanB}{1 – TanA.TanB}

Tan(A-B) ={TanA – TanB}{1 + TanA.TanB}

Cot(A+B) ={CotA CotB – 1}{CotA + CotB}

Cot(A-B) = {CotA CotB + 1}{CotB – CotA}

Sin2A = 2SinA. CosA = {2 TanA}{1 + Tan^2 A}

Cos2A = Cos²A – Sin²A =2Cos²A -1 = 1 – 2Sin²A ={1 – Tan^2 A}{1 + Tan ^2 A}

Tan2A ={2TanA}{1 – Tan^2 A}

Sin3A = 3SinA – 4Sin³ A

Cos3A = 4Cos³ A – 3 CosA

Tan3A ={3TanA -Tan^3 A}{1 – Tan^2 A}

Sin(-A) = SinA

Cos(-A) = – CosA

Sin0 = 0

Cos0 = 1

Sin(90 -A) = CosA, Cos(90-A) = SinA, Tan(90 – A) = CotA

Measure of angles of a trigonometric functions.