Trigonometric Functions

Trigonometric Functions, often simply called trig functions, are mathematical functions that relate the angles of a right triangle to the ratios of the lengths of its sides.

Trigonometric functions are the basic functions used in trigonometry and they are used for solving various types of problems in physics, Astronomy, Probability, and other branches of science. There are six basic trigonometric functions used in Trigonometry which are:

• Sine Function (sin x)
• Cosine Function (cos x)
• Secant Function (tan x)
• Cosecant Function (cosec x)
• Tangent Function (tan x)
• Cotangent Function (cot x)

Six Trigonometric Functions

The image added below shows a right-angle triangle PQR.

Then the six basic trigonometric functions formulas for this right angle triangle are,

Read More: Trigonometric function Ratios

Values of Trigonometric Functions

The value of trigonometric functions can easily be given using the trigonometry table. These values of the trigonometric functions are very useful in solving various trigonometric problems. The required trigonometry table is added below:

The table added above shows all the values of the important angles from 0 to 180 degrees for all the trigonometric functions.

Trigonometric Functions in Four(4) Quadrants

The trigonometric functions are the periodic functions and their values repeat after a certain interval. Also, not all the trigonometric functions are positive in all the quadrants.

An image explaining the same is added below:

We divide the cartesian space into four quadrants namely, I, II, III, and IV quadrants, and the value of the trigonometric functions whether they are positive or negative in each quadrant is given as,

• I Quadrant: All Positive
• II Quadrant: sin θ and cosec θ Positive
• III Quadrant: tan θ and cot θ Positive
• IV Quadrant: cos θ and sec θ Positive

Trigonometric Functions Graph

Trigonometric functions graphs plot the value of the trigonometric functions for different values of the angle(θ). For some the trigonometric functions are bounded as,

• Trigonometric functions sin θ and cos θ are bounded between - 1 and 1 and their graphs oscillate between -1 and 1 on the y-axis.
• Graph of the trigonometric function tan θ, and cot θ has a range from negative infinity to positive infinity.
• Graph of the trigonometric function sec θ, and cosec θ has a range from negative infinity to positive infinity excluding (-1, 1).

Read More: Graph of Trigonometric Functions

Domain and Range of Trigonometric Functions

Suppose we have a trigonometric function f(x) = sin x, then the domain of the function f(x) is all the values of x that the function f(x) can take, and the domain is all possible outcomes of the f(x). The domain and range of all the six trigonometric functions are:

Read in Detail- Domain and Range of Trigonometric Functions.

Properties of Trigonometric Functions

Some of the common properties of trigonometric functions are discussed below:

Period refers to the length of one complete cycle of a trigonometric function, after which the function repeats.

• Sine (sin), Cosine (cos), Secant (sec), Cosecant (csc): Period = 2π
• Tangent (tan), Cotangent (cot): Period = π

Symmetry refers to the property that describes how the function behaves under reflection, translation, or rotation.

• Even Functions: f(−θ) = f(θ) (Cosine and Secant).
• Odd Functions: f(−θ) = −f(θ) (Sine, Tangent, Cosecant, Cotangent).

Derivatives of Trigonometric Functions

Differentiation of trigonometric function can be easily found and the slope of that curve for that specific value of the trigonometric functions. The differentiation of all six trigonometric functions is added below:

• d/dx (sin x) = cos x
• d/dx (cos x) = -sin x
• d/dx (tan x) = sec²x
• d/dx (cot x) = -cosec²x
• d/dx (sec x) = sec x tan x
• d/dx (cosec x) = -cosec x cot x

Integration of Trigonometric Functions

As the integration of any curve gives the area under the curve, the integration of the trigonometric function also gives the area under the trigonometric function. The integration of various trigonometric functions is added below.

• ∫ cos x dx = sin x + C
• ∫ sin x dx = -cos x + C
• ∫ tan x dx = log|sec x| + C
• ∫ cot x dx = log|sin x| + C
• ∫ sec x dx = log|sec x + tan x| + C
• ∫ cosec x dx = log|cosec x - cot x| + C

Some other important trigonometric integrals are:

• ∫ sec²x dx = tan x + C
• ∫ cosec²x dx = -cot x + C
• ∫ sec x tan x dx = sec x + C
• ∫ cosec x cot x dx = -cosec x + C

Related Reads:,

• Application of Trigonometry in Real Life
• Trigonometric Equations
• Trigonometric Symbols