Project on Trigonometry for Class 10

The word "Trigonometry" is derived from three Greek words----' tri ' means three, 'gonia' means angle, and 'metron' means measure.Thus "Trigonometry" means  "three angle measure". Trigonometry deals with the relation between the angles and sides in a triangle.The study of trigonometry is of great importance in several fields and applied in many branches of  Science and Engineering such as Seismology,design of electrical circuits,estimating the heights of tides in the ocean etc.The three main trigonometric functions are sine,cosine and tangent and their reciprocals are co secant,secant and cotangent respectively.

Definitions of Trigonometric functions:

Definitions of Trigonometric Functions :-

Consider a right angle triangle and Let P(x,y) be the point and θ be the acute angle.

ON = x ; NP = y ; OP = r



The sine function is defined as ratio of opposite side (y) to hypotenuse(r).
sinθ = `(y)/(r)`

The cosine function is defined as ratio of adjacent side (x) to hypotenuse(r).
cosθ = `(x)/(r)`

The Tangent  function is defined as ratio of opposite side (y) to adjacent side (x).
tanθ = `(y)/(x)`

The reciprocal of sine is co secant and is defined as
cscθ = `(r)/(y)`

The reciprocal of cosine is secant and is defined as
secθ = `(r)/(x)`

The reciprocal of tangent is cotangent and is defined as
cotθ = x/y

Note:

sinθ and cscθ are reciprocal => sinθ * cscθ = 1
cosθand secθ are reciprocal => cosθ * secθ = 1
tanθ and cotθ are reciprocal => tanθ *  cotθ = 1


Formulas and Identities of Trigonometric functions


Identities :

sin2θ + cos2θ = 1
1 + tan2 θ = sec2 θ
1+ cot2 θ = csc2 θ

Formulas

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) = `(tan A + tan B)/(1- tan A tan B)`
tan(A-B) = `(tan A - tan B)/(1+ tan A tan B)`

Signs of Trigonometric Functions:

The entire coordinate plane is divided into 4 quadrants and they are named in counter clockwise direction.Let P(x , y) be a point in the coordinate plane.



(1) If P lies in the 1st quadrant,  all the trigonometric functions are positive.

(2) If P lies in the 2nd quadrant,  sinθ, cscθ are positive and the others are negative.

(3) If P lies in the 3rd quadrant, tanθ, cotθ are positive and the others are negative.

(4) If P lies in the 4th quadrant, cosθ, secθ are positive and the others are negative.\