Click here for the previous basics of trigonometry.
Basics of trigonometryNow we will study the next part of trigonometry.
1) Fundamental Identities:
a) sin2 Θ + cos2 Θ = 1
b) sec2 Θ - tan2 Θ = 1
c) csc2 Θ - cot2 Θ = 1
Proof:
a) sin2 Θ + cos2 Θ = 1
Here,
1) x2 + y2 = r2 -----------------(1)
Dividing equation 1 by r2 , we get,
2) (x2 / r2 + y2 / r2 = r2 / r2
3) (x / r) 2 + (y / r) 2 = 1 -------(2)
4) We know that,
y/r = sin Θ
x/r = cos Θ, ----- from equation 2, we have,
5) (cos Θ) 2 + (sin Θ) 2 = 1 ---(3)
6) (sin Θ) 2 + (cos Θ) 2 = 1
b) sec2 Θ - tan2 Θ = 1
Here,
1) x2 + y2 = r2 -------------------------(1)
Dividing equation 1 by x2,we get,
2) (x2 / x2 + y2 / x2 = r2 / x2
3) (x / x) 2 + (y / x) 2 = (r / x) 2 -------(2)
4) We know that,
y/x = tan Θ
r/x = sec Θ, ----- from equation 2, we have,
5) (1) 2 + (tan Θ) 2 = (sec Θ) 2 ---(3)
6) (sec Θ) 2 - (tan Θ) 2 = 1
c) csc2 Θ - cot2 Θ = 1
Here,
1) x2 + y2 = r2 -------------------------(1)
Dividing equation 1 by y2,we get,
2) (x2 / y2 + y2 / y2 = r2 / y2
3) (x / y) 2 + (y / y) 2 = (r / y) 2 -------(2)
4) We know that,
x/y = cot Θ
r/y = csc Θ, ----- from equation 2, we have,
5) (cot Θ) 2 + (1) 2 = (csc Θ) 2 ---(3)
6) (csc Θ) 2 - (cot Θ) 2 = 1
