Tiêu đề 39 Trigonometric Equations.pdf ✅

MSTC 39: Trigonometric Equations 1. Periodicity of Functions Consider the sine and cosine functions y = A sin[B(x + C)] + D y = A cos[B(x + C)] + D Shown are the graphs of sin x and cos x, Note that portions within 2π intervals. This repeating behavior is called periodic. The periods T of the sine and cosine (similarly, their reciprocals) are T = 2π B The heights of the waves increase when the value of A is increased. The heights of the waves get smaller when the value of A is decreased. The value of A is called the amplitude of the function. The waves move to the right when the value of C is increased. The waves move to the left when the value of C is decreased. The value of C is called the phase shift of the function. The waves move upward when the value of D is increased. The waves move downward when the value of D is decreased. The value of D is called the vertical shift of the function. Consider the tangent and cotangent functions y = A tan[B(x + C)] + D y = A cot[B(x + C)] + D These functions are periodic about π intervals. Their periods are T = π B The waves move to the right when the value of C is increased. The waves move to the left when the value of C is decreased. The value of C is called the phase shift of the function. The waves move upward when the value of D is increased. The waves move downward when the value of D is decreased. The value of D is called the vertical shift of the function. y = sin x y = cos x
Shown are the graphs of tan x and cot x, Consider the secant and cosecant functions y = A sec[B(x + C)] + D y = A csc[B(x + C)] + D Shown are the graphs of sec x and csc x, These functions are periodic about 2π intervals. Their periods are T = 2π B The waves move to the right when the value of C is increased. The waves move to the left when the value of C is decreased. The value of C is called the phase shift of the function. The waves move upward when the value of D is increased. The waves move downward when the value of D is decreased. The value of D is called the vertical shift of the function. Since the previous four functions are asymptotic whenever the denominator equates to zero (see quotient and reciprocal identities in MSTC 38: Trigonometric Equations), they have no amplitudes. y = tan x y = cot x y = sec x y = csc x
2. Trigonometric Equations Trigonometric equations are solved in the same manner as algebraic equations. However, due to their periodicity, the solutions are not unique. Consider the equation sin x = 1 Taking the inverse of both sides, Arcsin sin x = Arcsin 1 x = π 2 (or 90°) The value of x is directly computed with calculators. However, calculators can only output one value at a time. Often, they show the principal solutions to the equations. All angles coterminal to the principal value at periodic intervals should be added to account for the periodicity of the sine function. x = π 2 , 5π 2 , 9π 2 , ... A more compact way of writing this is x = π 2 ± 2πk ; k ∈ Z where k ∈ Z means that the value of k is any integer. Consider a compound trigonometric function, y = sin 2x + sin 3x The periods of the terms are π and 2π 3 , respectively. When a trigonometric function is composed of multiple parts, the overall period is equal to the least common multiple of the periods of each term. Taking the multiples of π and 2π 3 , π ⇒ π, 2π, 3π, ... 2π 3 ⇒ 2π 3 , 4π 3 , 2π, ... The least common multiple is 2π, which means that is the overall period of the function. This is called the fundamental period of the function. 3. Frequency The frequency of a function is the number of times the function repeats within a natural interval. It is the reciprocal of the period. Similarly, the fundamental frequency is the reciprocal of the fundamental period of a compound function.