Grade 12 – Advanced 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.