Problem 130 - 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) \(g(-\boldsymbol{\theta})\) $$\theta=-5 \pi / 3$$

Step-by-Step Solution

Answer

The answers are: (a) \(\sqrt{3}/2 + 1/2\), (b) \(1/2 - \sqrt{3}/2\), (c) \((1/2)^2\), (d) \(\sqrt{3}/2 * 1/2\), (e) \(\sqrt{3}/2\), and (f) \(1/2\).

Step 1: Converting Radians to Full Circle

First, convert the given angle \(\theta = -5\pi/3\) to an angle inside the unit circle. This is done by adding \(2\pi\) to the angle until it's between \(0\) and \(2\pi\). Our angle will be \(\theta = \pi/3\).

Step 2: Evaluate \(f(\theta)\) and \(g(\theta)\)

Next, evaluate \(f(\theta) = \sin \theta\) and \(g(\theta) = \cos \theta\) for \(\theta = \pi/3\). From the unit circle we know \(\sin \pi/3 = \sqrt{3}/2\) and \(\cos \pi/3 = 1/2\).

Step 3: Calculate (f+g)(Theta)

Add the values of \(f\) and \(g\), yielding \((f+g)(\theta) = \sqrt{3}/2 + 1/2\).

Step 4: Calculate (g-f)(Theta)

Subtract the value of \(f\) from \(g\), yielding \((g-f)(\theta) = 1/2 - \sqrt{3}/2\).

Step 5: Calculate \([g(\theta)]^{2}\)

Square the value of \(g\), yielding \([g(\theta)]^{2} = (1/2)^2\).

Step 6: Calculate (f g)(Theta)

Multiply the values of \(f\) and \(g\), yielding \((f g)(\theta) = \sqrt{3}/2 * 1/2\).

Step 7: Calculate f(2 Theta)

Given that twice Theta is \(\pi/3 * 2 = 2\pi/3\), look into the unit circle gives \(f(2\theta)= \sin 2\pi/3 = \sqrt{3}/2\).

Step 8: Calculate g(-Theta)

Given that minus Theta is \(- \pi/3\), look into the unit circle gives \(g(-\theta)= \cos -\pi/3 = 1/2\).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides, especially right-angled triangles. In the context of the unit circle, however, they become functions of an angle without the need to refer to a triangle.

There are six fundamental trigonometric functions, but for our exact trigonometric values exercise, we are focusing on sine (denoted as \(sin\)) and cosine (denoted as \(cos\)). Sine of an angle gives the ratio of the opposite side to the hypotenuse in a right-angled triangle, while cosine gives the ratio of the adjacent side to the hypotenuse. But when we refer to these functions in terms of the unit circle, they represent the y-coordinate and x-coordinate, respectively, of a point on the circle corresponding to an angle measured from the positive x-axis.

Unit Circle

The unit circle is a foundational concept in trigonometry. It's a circle with a radius of 1 centered at the origin (0,0) of a coordinate plane. It's incredibly useful because it allows us to define the trigonometric functions sine and cosine as functions of an angle, thus, making the functions relevant in a wide range of applications beyond triangles.

Any point on the unit circle has coordinates \( (cos\theta, sin\theta) \) where \(\theta\) is the angle in radians between the point and the positive x-axis. This relationship is a crucial part of solving trigonometric problems because it provides an exact geometric interpretation of the sine and cosine functions.

Radian Measure

Radian measure is an alternative way of expressing angles instead of the more familiar degrees. One radian is the angle created when the radius is wrapped along the circumference of a circle. It's crucial in trigonometry as it offers a more direct relationship to the unit circle.

There are \(2\pi\) radians in a full circle, thus dividing the circumference of the circle (which is \(2\pi r\), where \(r\) is the radius) by the radius. This gives us a very elegant and proportional relationship between the length of an arc of a circle and the angle that subtends it. In the case of the unit circle (\(r = 1\)), the arc length is equal to the radian measure of the angle subtending the arc.

Sine and Cosine

Sine and cosine are the two most widely used trigonometric functions and they form the basis for defining all other trigonometric functions. As noted earlier, these functions can be understood by looking at a right-angled triangle or the unit circle.

In our exercise, we calculate these functions at a specific angle, \(\theta\), using their relationship with the unit circle. For example, when \(\theta = \pi/3\), we find that \(f(\theta) = sin(\pi/3) = \sqrt{3}/2\) and \(g(\theta) = cos(\pi/3) = 1/2\). This relationship remains consistent for any angle and is a powerful tool in both pure and applied mathematics. By understanding how sine and cosine relate to the unit circle, students can more easily visualize and thus memorize the exact values for common angles.