Lesson 5 of 13 · 8 min
The other four trigonometric functions
NCERT §3.3
How steep is the spoke to Meera's cabin? Its slope is the rise over the run, sin x over cos x. Dividing sin and cos by each other, or into 1, gives four more functions.
The lesson in notes
In short
The remaining four functions are built from sine and cosine: tan x = sin x/cos x and sec x = 1/cos x, both defined only when x ≠ (2n + 1)π/2, n ∈ Z.
cot x = cos x/sin x and cosec x = 1/sin x are defined only when x ≠ nπ, n ∈ Z, because sin x is zero there.
So cosec x, sec x and cot x are the reciprocals of sin x, cos x and tan x, wherever both functions in the pair are defined.
Dividing cos² x + sin² x = 1 by cos² x gives 1 + tan² x = sec² x, and dividing it by sin² x gives 1 + cot² x = cosec² x, each valid wherever its terms are defined.
Values at the standard angles 0, π/6, π/4, π/3, π/2: sin takes 0, 1/2, 1/√2, √3/2, 1; cos takes 1, √3/2, 1/√2, 1/2, 0; tan takes 0, 1/√3, 1, √3, and is not defined at π/2.
The reciprocal functions follow from these: for example, cosec π/6 = 2, sec π/4 = √2 and cot π/3 = 1/√3, while cot 0 and cosec 0 are not defined.
Tan and cot can be written through the other functions: tan x = sin x × sec x, and sin x = tan x × cos x, which is often the quickest route from one given value to another.