Trigonometric Functions Graphs of Sine, Cosine, and Tangent ...
Trigonometric Functions Graphs of Sine, Cosine, and Tangent Functions
The graphs of y = sin x, y = cos x, and y = tan x are periodic.
The graphs of y = sin x and y = cos x are similar in shape and have an amplitude of 1 and a period of 2π
The graph of y = sin x can be transformed into graphs modeled by equations of the form y = sin x + c, y = sin (x – d), and y = sin kx. Similarly, the graph of y = cos x can be transformed into graphs modeled by equations of the form y = cos x + c, y = acos x, y = cos (x – d), and y = cos kx.
The graph of y = tan x has no amplitude because it has no maximum or minimum values. It is undefined at values of x that are odd multiples of π/2, such as π/2 and 3π/2.
The graph becomes asymptotic as the angle approaches these values from left and the right. The period of the function is π.
Graphs of Reciprocal Trigonometric Functions
The graphs of y = csc x, y = sec x, and y = cot x are periodic. They are related to the primary trigonometric functions as reciprocal graphs.
Reciprocal trigonometric functions are different from inverse trigonometric functions.
csc x means 1 / sin x, while sin-1 x asks you to find an angle that has a sine ratio equal to x.
sec x means 1 / cos x, while cos-1 x asks you to find an angle that has a cosine ratio equal to x.
cot x means 1 / tan x, while tan-1 x asks you to find an angle that has a tangent ratio equal to x.
Sinusoidal functions of the form f(x) = a sin[k(x - d)] + c and f(x) = a cos[k(x - d)] + c
The transformation of a sine or cosine function f(x) to g(x) has the general form g(x) = a f [k(x - d)] + c, where |a| is the amplitude, d is the phase shift, and c is the vertical translation.
The period of the transformed function is given by 2π / k.
The k value of the function is given by 2π / period.
Solve Trigonometric Equations
Trigonometric equations can be solved algebraically by hand or graphically using technology.
There are often multiple solutions. Ensure that you find all solutions that lie in the domain of interest.
Quadratic trigonometric equations can often be solved by factoring.
Often, a trigonometric equation might need to be manipulated using trigonometric identities in order of it to be solved. Refer to notes on trigonometric identities here.
Instantaneous Rate of Change Application
The instantaneous rates of change of a sinusoidal function follows a sinusoidal pattern.
Without the knowledge of limits, a gradual substitution of a number closer and closer to the expected value will determine the instantaneous rate of change.
The graphs of y = sin x, y = cos x, and y = tan x are periodic.
The graphs of y = sin x and y = cos x are similar in shape and have an amplitude of 1 and a period of 2π
The graph of y = sin x can be transformed into graphs modeled by equations of the form y = sin x + c, y = sin (x – d), and y = sin kx. Similarly, the graph of y = cos x can be transformed into graphs modeled by equations of the form y = cos x + c, y = acos x, y = cos (x – d), and y = cos kx.
The graph of y = tan x has no amplitude because it has no maximum or minimum values. It is undefined at values of x that are odd multiples of π/2, such as π/2 and 3π/2.
The graph becomes asymptotic as the angle approaches these values from left and the right. The period of the function is π.
Graphs of Reciprocal Trigonometric Functions
The graphs of y = csc x, y = sec x, and y = cot x are periodic. They are related to the primary trigonometric functions as reciprocal graphs.
Reciprocal trigonometric functions are different from inverse trigonometric functions.
csc x means 1 / sin x, while sin-1 x asks you to find an angle that has a sine ratio equal to x.
sec x means 1 / cos x, while cos-1 x asks you to find an angle that has a cosine ratio equal to x.
cot x means 1 / tan x, while tan-1 x asks you to find an angle that has a tangent ratio equal to x.
Sinusoidal functions of the form f(x) = a sin[k(x - d)] + c and f(x) = a cos[k(x - d)] + c
The transformation of a sine or cosine function f(x) to g(x) has the general form g(x) = a f [k(x - d)] + c, where |a| is the amplitude, d is the phase shift, and c is the vertical translation.
The period of the transformed function is given by 2π / k.
The k value of the function is given by 2π / period.
Solve Trigonometric Equations
Trigonometric equations can be solved algebraically by hand or graphically using technology.
There are often multiple solutions. Ensure that you find all solutions that lie in the domain of interest.
Quadratic trigonometric equations can often be solved by factoring.
Often, a trigonometric equation might need to be manipulated using trigonometric identities in order of it to be solved. Refer to notes on trigonometric identities here.
Instantaneous Rate of Change Application
The instantaneous rates of change of a sinusoidal function follows a sinusoidal pattern.
Without the knowledge of limits, a gradual substitution of a number closer and closer to the expected value will determine the instantaneous rate of change.
