Problem 121 - 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) \(g(-\boldsymbol{\theta})\) $$\theta=30^{\circ}$$

Step-by-Step Solution

Answer

\((f+g)(\theta) = (\sqrt{3}+1)/2, (g-f)(\theta) = (\sqrt{3}-1)/2, [g(\theta)]^{2} = 3/4, (f g)(\theta) = \sqrt{3}/4, f(2\theta) = \sqrt{3}/2, g(-\theta) = \sqrt{3}/2.

Step 1

Find the values for \(f(\theta)=\sin\theta\) and \(g(\theta)=\cos\theta\) for \(\theta=30^{\circ}\). Here \(f(30^{\circ}) = \sin(30^{\circ}) = 0.5\) and \(g(30^{\circ}) = \cos(30^{\circ}) = \sqrt{3}/2\).

Step 2

Calculate \((f+g)(\theta) = f(30^{\circ}) + g(30^{\circ}) = 0.5 + \sqrt{3}/2 = 1/2 + \sqrt{3}/2 = (\sqrt{3}+1)/2 \).

Step 3

Calculate \((g-f)(\theta) = g(30^{\circ}) - f(30^{\circ}) = \sqrt{3}/2 - 0.5 = \sqrt{3}/2 - 1/2 = (\sqrt{3}-1)/2 \).

Step 4

Compute \([g(\theta)]^{2} = [g(30^{\circ})]^2 = (\sqrt{3}/2)^2 = 3/4 \).

Step 5

Compute \((f g)(\theta) = f(30^{\circ}) \cdot g(30^{\circ}) = 0.5 \cdot \sqrt{3}/2 = \sqrt{3}/4\).

Step 6

Find \(f(2\theta) = \sin(2 \cdot 30^{\circ}) = \sin(60^{\circ}) = \sqrt{3}/2\).

Step 7

Finally calculate \(g(-\theta) = \cos(-30^{\circ}) = \cos(30^{\circ}) = \sqrt{3}/2.\)

Key Concepts

Sine and Cosine

Sine (\textbf{sin}) and cosine (\textbf{cos}) are foundational trigonometric functions that correspond to the ratios of sides of a right triangle relative to a specific angle.

For any angle \( \theta \), the sine function represents the ratio of the length of the side of the triangle opposite the angle (\( \theta \)) to the length of the hypotenuse. In the unit circle framework, sine gives the y-coordinate of a point on the circle's circumference. The formula for sine in a right triangle is \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).

Cosine, on the other hand, represents the ratio of the adjacent side's length to the hypotenuse's length. On the unit circle, the cosine function provides the x-coordinate. The formula is \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \). These functions are periodic, with sine and cosine having a period of \( 2\pi \) radians (or \( 360^\circ \) degrees). They are essential in calculating angles and distances in various fields such as physics, engineering, and geography.

Exact Values of Trigonometric Functions

The exact values of trigonometric functions for specific angle measures are critical in mathematics as they allow for precise calculations without the use of a calculator. These exact values are often found in special angles such as \(30^\circ\), \(45^\circ\), and \(60^\circ\) which correspond to common ratios in an equilateral and isosceles right triangle.

For example, \(\sin(30^\circ) = \frac{1}{2}\) and \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\), as shown in the solution to the given exercise. Memorizing the exact values for these key angles facilitates the understanding of more complex trigonometric concepts and the evaluation of trigonometric expressions. They are often represented in a tabulated form known as the 'unit circle' or derived from the properties of specific triangles, such as the 45-45-90 triangle (isosceles right triangle) or the 30-60-90 triangle (half of an equilateral triangle).

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the involved variables. These identities simplify complex trigonometric equations and aid in solving for unknown angles and lengths.

One fundamental identity is the Pythagorean identity, which states that for any angle \( \theta \), \( \sin^2(\theta) + \cos^2(\theta) = 1 \). This identity arises from the Pythagorean theorem applied to a unit circle and is pivotal in transforming trigonometric expressions.

Other common identities include angle sum and difference formulas, double and half-angle formulas—like \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)—and equations that denote \( \tan(\theta) \) as the ratio of \( \sin(\theta) \) to \( \cos(\theta) \). Knowing these identities is crucial for effectively navigating trigonometry problems like the exercise provided.

Angle Measures in Trigonometry

Angles in trigonometry are measured in degrees or radians. Degree measure is the traditional way of expressing angles, based on dividing a circle into 360 equal parts. Radian measure, the standard unit of angular measure used in many areas of mathematics, is based on the radius of a circle. A full revolution is \( 2\pi \) radians, which is equal to \( 360^\circ \).

The conversion between these two measurements is vital: \(1^\circ = \frac{\pi}{180}\) radians and conversely \(1\) radian \(= \frac{180}{\pi}\) degrees. When working with trigonometric functions, it is often necessary to convert angles from degrees to radians or vice versa to use certain identities or to integrate or differentiate trigonometric functions. In the exercise example, \( \theta = 30^\circ \) is used, but knowing how to work with radians could be necessary for more complex problems.