Trigonometric Functions Solutions

Trigonometric Functions Trigonometric functions are very useful in a number of applications. Sinusoidal (sine and cosine) functions are used to represent quantities that are oscillating with time (moving between a minimum and maximum value). Because we want to describe systems that are oscillating, we will focus on the sine and cosine functions. When studying calculus, it is customary to use radians rather than degrees. In addition to being unitless, the length of an arc on a unit circle is the same as the angle of the arc measured in radians. On the other hand, degrees are a unit of angle measure, and the definition of 360 degrees in a circle is rather arbitrary mathematically (as well as lacking in some nice properties that trigonmetric functions defined in terms of radians possess). Nevertheless, measurements are usually made in degrees, so we will want to convert such measurements into radians, which are mathematically simpler to deal with. The observation that there are 360◦ or 2π radians in a circle allows us to create a conversion factor between degrees and radians. From 180◦ = π radians it follows that 180◦ π radians and 1 = π radians 180◦ For instance, from the above equations we can see that 60◦ = π3 radians, and that 1= π 4 radians = 45◦ . The unit circle is a circle of radius 1 centered at the origin. The Cartesian coordinates of any point (x, y) on the unit circle are equal to (cos(θ), sin(θ)) where θ is the angle between the positive x-axis and the arc reaching the point (x, y). As functions of x, the sine and cosine functions look as follows If a function repeats itself after a given distance in its argument, then the function is said to be periodic, and the period is the distance before the function repeats. We can see that both sine and cosine repeat every 2π, so they are functions with periods of 2π. Mathematically this can be written as cos(θ) = cos(θ + 2nπ) sin(θ) = sin(θ + 2nπ) where n ∈ Z (n is an integer). We can also see that sin(x) and cos(x) are essentially the same function, except for that one is a shifted version of the other. In fact π ) 2 Thus, we can use the cosine function to calculate the value of a sine function, and vice-versa. We’ll make an arbitrary choice to use cosine rather than sine henceforth. sin(θ) = cos(θ − A general sinusoidal function constructed from a cosine can be written as follows f (t) = A + B · cos In the above equation
2π (t − φ) T • B is the amplitude of the function, which is half the difference of the maximum and minimum values • A is the average value of the function, which lies the amplitude below a maximum, or the amplitude above a mimimum • T is the period of the function, which is the distance between successive maxima (or minima) • φ is the phase of the function, which is the time of the first maximum From the above relationships it follows that max = A+B and min = A−B. Also, the first maximum is at t = φ, the first minimum is at φ + T2 , and the average value occurs at t = φ + T4 and t = φ + 3T 4 . Example 1 Plot g(t) = 1 + 0.5 · cos( π5 t − π). Solution The first step is to write this sinusoid in the our standard form. Thus 2π 2π 2π π 5) = 1 + 0.5 · cos( (t − 5)) g(t) = 1 + 0.5 · cos( t − π) = 1 + 0.5 · cos( t − 5 10 10 10 Now we can read that A = 1, B = 0.5, T = 10, and φ = 5. Since max = A + B, and occurs at t = φ, g(5) = 1.5. The next maximum occurs at t = 15, and two minima occur at t = 0 and t = 10. The function has its average values at t = 2.5, 7.5, and 12.5. 1.5 1.4 1.3 1.2 g(t) 1.1 1 0.9 0.8 0.7 0.6 0.5 -5 0 5 t 10 15