Problem 134 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=5 \pi / 2$$

Step-by-Step Solution

Answer

(a): 1, (b): -1, (c): 0, (d): 0, (e): 0, (f): 0

Step 1: Evaluate original functions

First, evaluate \( f(\theta)\) and \( g(\theta) \). Recall that the sine function returns to its original value every 2π, so \(f(\theta) = \sin(5 \pi / 2) = \sin(\pi / 2) = 1\). The cosine function also returns to its original value every 2π, so \(g(\theta) = \cos(5 \pi / 2) = \cos(\pi / 2) = 0\).

Step 2: Evaluate composite functions

Now, using the values from step 1, evaluate the composite functions. \n(a) \((f+g)(\theta) = f(\theta) + g(\theta) = 1 + 0 = 1\)\n(b) \((g-f)(\theta) = g(\theta) - f(\theta) = 0 - 1 = -1\)\n(c) \([g(\theta)]^{2} = [g(\theta)]^2 = 0^2 = 0\)\n(d) \((f g)(\theta) = f(\theta)g(\theta) = 1 * 0 = 0\).

Step 3: Evaluate scaled functions

Lastly, evaluate the functions where the argument of the function is scaled by a factor. \n(e) \(f(2 \theta) = \sin(2 * 5 \pi / 2) = \sin(5 \pi) = \sin( \pi) = 0\)\n(f) \(g(-\theta) = \cos(-5 \pi / 2) = \cos(- \pi / 2) = 0\). After understanding that the cosine function is even, which means \( \cos(-x) = \cos(x) \).

Key Concepts

Sine Function

The sine function, or simply 'sine', is a fundamental trigonometric function that describes the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. When considering the unit circle (a circle with a radius of 1), the sine of an angle is the y-coordinate of the corresponding point on the circle.

For example, in the step-by-step solution provided for the exercise, we start by evaluating the sine function at an angle of \( 5 \pi / 2 \). However, we recognize that the sine function has a period of \(2\pi\), so it repeats its values every \(2\pi\) radians. This means that we can simply take the sine of \( \pi / 2 \) to determine \( f(\theta) \). In this case, \( \sin(\pi / 2) = 1 \), as the point \( (0, 1) \) on the unit circle corresponds to this angle.

Cosine Function

The cosine function is another foundational trigonometric function that is crucial for understanding the relationships within a circle or triangle. It is defined as the ratio of the length of the adjacent side to the hypotenuse in a right angle triangle, and in terms of the unit circle, it's the x-coordinate of a point on the circle.

In the given solution, to determine \( g(\theta) \), we calculate the cosine of \( 5 \pi / 2 \). Similar to the sine function, the cosine function has a period of \(2\pi\) and will return to the same value after each full cycle. As a result, we compute \( \cos(\pi / 2) \) instead, obtaining a value of 0. This reflects the point \( (0, 0) \) at the top of the unit circle where the angle \( \pi / 2 \) is positioned.

Composite Trigonometric Functions

Composite trigonometric functions are created by combining two or more trigonometric functions or by applying operations to them. It is a more complex concept where different functions can be added, subtracted, multiplied, divided, or composed with each other in a single expression.

In our step-by-step solution, we perform operations with \( f(\theta) \) and \( g(\theta) \) by adding, subtracting, and multiplying them. For \( (f+g)(\theta) = \sin(\theta) + \cos(\theta) \), this means taking the sum of the sine and cosine values at the same angle. The expression for \( (g-f)(\theta) = \cos(\theta) - \sin(\theta) \) requires subtracting one function from another. As seen in our example, results can be as simple as 1, 0, or -1, but the process gives way to much richer analysis in more complex scenarios.

Angle Measurement in Radians

Radians are a unit of angle measurement that are based on the radius of a circle. One radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle. In trigonometry, radians are often more convenient than degrees for angle measurement, especially when working with trigonometric functions which have properties well-aligned with the radian measurement system.

For example, when evaluating trigonometric functions as we do in this exercise, recognizing that \(2\pi\) radians is one full rotation around the circle (equivalent to 360 degrees) simplifies the process. This is why we can convert \(5\pi/2\) into \(\pi/2\) when calculating the sine or cosine of the angle. Understanding radians is crucial because it connects angular motion to linear motion and allows for straightforward integration into calculus.