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4.3: Equivalent Trigonometric Expressions   

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4.3: Equivalent Trigonometric Expressions   

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Equivalent trigonometric expressions are expressions that simplify to the same value. Using Equivalent Expressions

Using the equality of expressions on two sides

Substituting known values to simplify an expression

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Expressing one trigonometric function in terms of the function constant for the other

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Expressing one trigonometric function in terms of the function constant for the other

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The trigonometric functions are taken out by arrangement of the equations first, and later, in the given intervals, the solutions are determined, usually with many solutions in different quadrants.

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Solve for angles using known values and inverse functions and determine all possible solutions in a given interval.

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Solve for angles using known values and inverse functions and determine all possible solutions in a given interval.

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Formula and Basic Form
The general formula for sinusoidal functions (sine and cosine) is:

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Parts of the Formula
Each part of the formula changes the way the graph looks:

Amplitude a: This shows how high and low the wave goes. The larger aaa is, the taller the wave.

Frequency k: This shows how many waves (or cycles) there are in a set amount of space. A larger kkk makes the wave repeat more often.

Phase Shift d: This moves the wave left or right. If d is positive, the wave moves to the right; if negative, it moves to the left.

Vertical Shift c: This moves the wave up or down on the graph. A positive c moves it up, and a negative c moves it down.

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Amplitude (a):

The value of ∣a∣ determines the height of the wave from its midline (the horizontal line defined by y=c).

If a>0, the sine or cosine wave opens upwards; if a<0, it opens downwards.

Example: For a=3, the amplitude is 3, meaning the function oscillates 3 units above and below the midline.

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Frequency (k):

The value of k affects the period of the function. The period T is calculated using the formula:

A larger value of ∣k∣ results in a shorter period (more oscillations in a given interval), while a smaller value results in a longer period.

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Phase Shift (d):

The value of d shifts the graph horizontally. The phase shift can be found by setting x−d=0, leading to a shift of d units to the right. If d is negative, it shifts to the left.

Example: For d=2, the graph shifts 2 units to the right.

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Vertical Shift (c):

The value of c shifts the entire graph vertically. The midline of the wave moves to y=c.

If c>0, the graph shifts upward; if c<0, it shifts downward.

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What is Instantaneous Rate of Change?

The instantaneous rate of change is how fast a function is changing at a specific point. For sinusoidal functions (like sine and cosine waves), this rate changes along the curve.

---The instantaneous rates of change of a sinusoidal function follow a sinusoidal pattern

---Many real-world situations can be modelled with a sinusoidal function, even if they do not involve

---Modelling real-world situations usually requires transformations of the basic sinusoidal functuins

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5.1 Graphs of the Sine, Cosine, and Tangent Functions

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5.1

Periodicity of function graphs: The graphs of y = sinx, y = cosx, and y = tanx are periodic.

1. Features of the sine and cosine functions: y = sinx and y = cosx have similar shapes, an amplitude of 1, and a period of 2π. 2.Transform the graphs of the sine and cosine functions:

y = sinx can be image transformed by the equations y = sinx + c, y = asinx, y = sin (x - d), and y = sinkx.

y = cosx can be transformed into an graphs in the form of the equations y = cosx + c, y = acosx, y = cos (x - d), and y = coskx.

4.Features of the tangent function: y = tanx has no amplitude (no maximum or minimum), is undefined at odd multiples of x for π/2 (e.g., π/2, 3π/2), and has an asymptotic graph as the angle approaches these values from the left and right, and the period of the function is π.

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5.2 Graphs of reciprocal trigonometric functions

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5.2

Periodicity of functions and relationships: the graphs of y = cscx, y = secx and y = cotx are periodic and they are reciprocal to the basic trigonometric graphs.

reciprocal trigonometric functions and inverse trigonometric functions they are different.

Specific function meanings:

cscx is 1 /sinx, and sin-¹x is to find the angle whose sine is equal to x.

secx is 1 /cosx, and cos-¹x is the angle whose cosine is equal to x.

cotx is 1 /tanx, and tan-¹x is the angle whose tangent is equal to x.