The six fundamental trigonometric functions sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec), and cotangent (cot) each have unique properties such as periodicity, symmetry, and specific domain and range restrictions.
Understanding these properties helps solve equations and simplify expressions.
Periodicity
Trigonometric functions are periodic, meaning they repeat their values at regular intervals.
Sine, Cosine, and Tangent
Both sine and cosine functions have a period of 2π, whereas the tangent function has a period of π, which implies:
sin(x + 2π) = sin x
cos(x + 2π) = cos x
tan(x + π) = tan x
Secant, Cosecant, and Cotangent
Similarly, the periods of the reciprocal functions (secant, cosecant, and cotangent) are:
sec(x + 2π) = sec x
cosec(x + 2π) = cosec x
cot(x + π) = cot x
Symmetry
The symmetry of trigonometric functions determines whether they are even or odd, which simplifies calculations in integrals and derivatives and helps analyze their graphs.
Even Functions
The cosine and secant are even functions because their values remain the same for opposite angles:
cos(-x) = cos x
sec(-x) = sec x
Odd Functions
The sine, tangent, cosecant, and cotangent are odd functions because their signs reverse for opposite angles:
sin(-x) = -sin x
tan(-x) = -tan x
cot(-x) = -cot x
cosec(-x) = -cosec x
Domain and Range
Each trigonometric function has distinct restrictions on its inputs (domain) and outputs (range), primarily due to its periodic and geometric nature.
Sine and Cosine
Domain: All real numbers (ℝ)
Range: [-1, 1]
Tangent and Cotangent
Domain: All real numbers except where the function is undefined. The domain of tan x is: ${x\in \mathbb{R} -\left\{ \dfrac{\pi }{2}+n\pi | n\in \mathbb{Z} \right\}}$ and the domain of cot x is: ${x\in \mathbb{R} -\left\{ n\pi | n\in \mathbb{Z} \right\}}$
Range: All real numbers (ℝ)
Secant and Cosecant
Domain: All real numbers except where sin x = 0 (for cosec x) and cos x = 0 (for sec x). The domain of sec x is: ${x\in \mathbb{R} -\left\{ \dfrac{\pi }{2}+n\pi | n\in \mathbb{Z} \right\}}$ and the domain of cosec x is: ${x\in \mathbb{R} -\left\{ n\pi | n\in \mathbb{Z} \right\}}$
Range: (-∞, -1] ∪ [1, ∞)
Asymptotes
Certain trigonometric functions have vertical asymptotes where they are undefined:
Tangent: Vertical asymptotes at x = ${\dfrac{\pi }{2}+n\pi}$
Cotangent: Vertical asymptotes at x = nπ
Secant: Vertical asymptotes where cos x = 0
Cosecant: Vertical asymptotes where sin x = 0
Continuity
The sine and cosine graphs are smooth and continuous. It oscillates between -1 and 1.
The tangent and cotangent graphs are periodic, with vertical asymptotes and repeating patterns.
The secant and cosecant graphs are discontinuous with vertical asymptotes and arcs.
From Graph
We can also find the above properties from the graphs of the respective trigonometric functions.
Sine Function
Sine Graph Unit Circle
Period: 2π
Type of function: Odd function
Domain: (-∞, ∞)
Range: [-1, +1]
Amplitude: 1
X-Intercept: x = nπ, for all n
Y-Intercept: y = 0
Line of symmetry: At the origin (0, 0)
Cosine Function
Cosine Graph Unit Circle
Period: 2π
Type of function: Even function
Domain: (-∞, ∞)
Range: [-1, +1]
Amplitude: 1
X-Intercept: x = ${\left( 2n+1\right) \dfrac{\pi }{2}}$, for all n
Vertical asymptotes: x = ${\left( 2n+1\right) \dfrac{\pi }{2}}$
Cosecant Function
Period: 2π
Type of function: Odd function
Domain: ℝ – nπ
Range: (-∞, -1] ∪ [1, ∞)
X-Intercept: It does not exist
Y-Intercept: It does not exist
Line of symmetry: At the origin (0, 0)
Vertical asymptotes: x = nπ
From Identities
We can also get to the properties of trigonometric functions using specific trigonometric identities. Here is the list of those identities:
For example,
Using the opposite angles identity, sin(-x) = -sin x, which confirms that sine is an odd function.
Using the periodic identity, sin(2nπ + θ) = sin θ, which confirms that sin θ has a period of 2π
Using the reciprocal identity, ${\tan \theta =\dfrac{1}{\cot \theta }}$, which confirms that tan θ and cot θ are reciprocal.
Solved Examples
Simplify cos2 x – sin2 x using trigonometric identities and verify its periodicity.
Solution:
As we know, the double-angle identity is: cos2 x – sin2 x = cos 2x The period of cos 2x is 2π, meaning cos(x + 2π) = cos x For cos 2x, the function completes one full cycle in half the interval of cos x Hence, the period of cos 2x is: Period = ${\dfrac{2\pi }{2}}$ = ${\pi}$ This confirms that cos 2x is periodic with a period of π Thus, the simplified form of cos2 x – sin2 x is cos 2x, and it has a period of π
Verify whether sin x and cosec x are reciprocal functions by evaluating their product at x = ${\dfrac{\pi }{4}}$
Solution:
As we know, the reciprocal relationship between sin x = ${\dfrac{1}{\text{cosec}\, x}}$ Substituting x = ${\dfrac{\pi }{4}}$, we get ${\sin \dfrac{\pi }{4}=\dfrac{1}{\sqrt{2}}}$ ${\text{cosec}\, \dfrac{\pi }{4}=\dfrac{1}{\text{cosec}\, \dfrac{\pi }{4}}\cdot \text{cosec}\, \dfrac{\pi }{4}=\sqrt{2}}$ Now, at x = ${\dfrac{\pi }{4}}$, sin x ⋅ cosec x = ${\dfrac{1}{\text{cosec}\, \dfrac{\pi }{4}}\cdot \text{cosec}\, \dfrac{\pi }{4}}$ = ${\dfrac{1}{\sqrt{2}}\cdot \sqrt{2}}$ = 1 ⇒ sin x ⋅ cosec x = 1, at x = ${\dfrac{\pi }{4}}$ Thus, sin x and cosec x are reciprocal functions.
