Trigonometric Functions

There are six trigonometric functions: Sine function, Cosine function, Tangent function, Cosecant function, Secant function and Cotangent function.

1. Sine function

fx=sinsin x  is called sine function.

      Graph of Sine function:

Domain and Range of Sine function:

Domain of y=sinsin x  is R and Range is [-1,1].

            Properties of Sine function:

• sinsin x  is a periodic function having period 2π,i.e., graph of sine function repeats after regular interval of length 2π units, so, y=sinsin x  is a many-one function.
• Graph of y=sinsin x  is symmetric about origin, so, sine function is called odd function.
• Sine function is bounded function as range of sine function lies between -1 to 1.

2. Cosine function

fx=coscos x  is called cosine function.

Graph of Cosine function:

Domain and Range of Cosine function:

• Domain of y=coscos x  is R and Range is [-1,1].

Properties of Cosine function:

• coscos x  is a periodic function having period 2π,i.e., graph of cosine function repeats after regular interval of length 2π units, so, y=coscos x  is many-one function.
• Graph of y=coscos x  is symmetric about y-axis, so, cosine function is an even function.
• Cosine function is bounded function as range of cosine function lies between -1 to 1.

3. Tangent function

fx=tantan x  is called tangent function.

Graph of Tangent function:

Domain and Range of tangent function:

• Domain of y=tantan x  is R-2n+1π2, n∈Z and Range is R.

Properties of tangent function:

• tantan x  is a periodic function having period π,i.e., graph of tangent function repeats after regular interval of length π units, so, y=tantan x  is a many-one function.
• Graph of y=tantan x  is symmetric about origin, so, tangent function is an odd function.
• Tangent function is unbounded function as its range lies between -∞ to ∞.

4. Cosecant function

fx=x  is called cosecant function.

Graph of Cosecant function:

 Domain and Range of Cosecant function:

• Domain of y=x  is R-nπ,n∈Z and Range is R-(-1,1).

Properties of Cosecant function:

• Cosecant function & Sine function are reciprocal of each other ⇒ cosec x=1sinsin x
• x  is a periodic function having period 2π,i.e., graph of cosecant function repeats after regular interval of length 2π units, so, y=x  is a many-one function.
• Graph of y=x  is symmetric about origin, so, cosecant function is called odd function.
• Cosecant function is unbounded function as it can take any values except values lying between -1 to 1.

5. Secant function

fx=secsec x  is called secant function.

Domain and Range of Secant function:

• Domain of y=secsec x  is R-2n+1π2,n∈Z and Range is R-(-1,1).

Properties of Secant function:

• Secant function & cosine function are reciprocal of each other ⇒ secsec x =1coscos x
• secsec x  is a periodic function having period 2π,i.e., graph of secant function repeats after a regular interval of length  2π units, so, y=secsec x  is a many-one function.
• Graph of y=secsec x  is symmetric about y-axis, so, secant function is an even function.
• Secant function is unbounded function as it can take any values except values lying between -1 to 1.

6. Cotangent function

fx=cotcot x  is called cotangent function.

Graph of Cotangent function:

Domain & Range of Cotangent function:

• Domain of y=cotcot x  is R-nπ,n∈Z and Range is R.

Properties of Cotangent function:

• Cotangent function & tangent function are reciprocal of each other ⇒ cot x=1tantan x .
• cotcot x  is a periodic function having period π,i.e., graph of cotangent function repeats after a regular interval of length  π units, so, y=cotcot x  is a many-one function.
• Graph of y=cotcot x  is symmetric about origin, so, cotangent function is an odd function.
• Cotangent function is unbounded function as it can take any value lying between -∞ to ∞.

Solved Examples:

Example 1:

Find the domain of  11-2sinsin x

Solution:

11-2sinsin x  is defined if 1-2sinsin x≠0

⇒sinsin x ≠12

⇒sinsin x ≠sinsin π6

⇒x≠nπ+-1nπ6, n∈Z

Hence, the domain is R-nπ+-1nπ6 

Example 2:

Find the range of fx=12 cos x-1

Solution:

⸪  -1≤cos x≤1

⇒-2≤2coscos x≤2

⇒-3≤2coscos x-1≤1

⇒-3≤2coscos x-1<0  or 0<2coscos x-1≤1

⇒-∞<12coscos x -1≤-13 or 1≤12coscos x -1<∞

Hence, the range of f(x) is ]-∞,-13]∪[1,∞[.

Example 3: Find domain of fx=sin x+16-x2

Solution:

For fx=sin x+16-x2 to be defined if sinsin x  and 16-x2 is defined.

16-x2 is defined when 16-x2≥0

⇒x2≤16

⇒x∈[-4,4]                           (⸪x2≤a2⇒x∈[-a,a])

Now, sinsin x  is defined if sinsin x ≥0

Clearly, sinsin x ≥0, ∀x∈-4,-π∪[0,π] in the interval x∈[-4,4].

Hence, the domain of fx is x∈-4,-π∪[0,π]

FAQs

1. What are the points where y=tan x is not defined.

Solution:

Since, the domain of y=tantan x  is R-2n+1π2, n∈Z

So, y=tantan x  is not defined at odd multiples of π2.

2. What are the points where y=cot x is not defined.

Solution:

Since, the domain of y=cotcot x  is R-nπ, n∈Z

So, y=cotcot x  is not defined at integral multiples of π.