Problem 17 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

\(g(x)=3 \csc x\)

Step-by-Step Solution

Answer

The simplified form of the function is \(g(x) = 3 / \sin x\)

Step 1: Identify the function

The function given is \(g(x)=3 \csc x\). Here, \(g(x)\) can be any real number while \(x\) is an angle.

Step 2: Understand the identity of csc

Trigonometric functions include sine, cosine, tangent, cotangent, secant, and cosecant. Often, to simplify, we find it helpful to express the cotangent, secant, and cosecant in terms of sine, cosine, and tangent. Remember that csc x = 1/sin x.

Step 3: Substitute csc x with 1/sin x

Now, replacing csc x in our function with 1/sin x gives us a simpler version of our function as follows: \(g(x) = 3 * 1/\sin x \).

Step 4: Simplify the function

This simplification leads to: \(g(x) = 3 / \sin x\).

Key Concepts

Cosecant

The cosecant function, denoted as \( \csc x \), is one of the six fundamental trigonometric functions. It is the reciprocal of the sine function. This means:

• \( \csc x = \frac{1}{\sin x} \)

Understanding cosecant is helpful because it completes the trigonometric family along with sine, cosine, tangent, secant, and cotangent. Cosecant is particularly useful in simplifying trigonometric expressions and solving equations.
To conceptualize \( \csc x \), visualize a right triangle: if \( \sin x \) is the ratio of the opposite side over the hypotenuse, \( \csc x \) represents the hypotenuse over the opposite side. Therefore, wherever the sine is zero, cosecant tends to infinity, signifying its undefined nature at those points.
When working with trigonometric functions, always remember that the key to simplifying them often involves knowing reciprocal identities such as that of cosecant.

Sine

The sine function, represented as \( \sin x \), is one of the primary trigonometric functions. It is defined as the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. In the unit circle, sine is the y-coordinate of the point where the terminal side of an angle intersects the circle.

• \( \sin x = \frac{\text{Opposite}}{\text{Hypotenuse}} \)

Sine is periodic, with a period of \( 2\pi \), meaning it repeats its values in regular intervals. This function is fundamental in describing waves and oscillations, affecting fields such as physics, engineering, and even music.
One of the intriguing properties of sine is its symmetry: \( \sin(-x) = -\sin(x) \), which makes it an odd function. This trait helps in analyzing and simplifying trigonometric equations and signals. Recognizing these patterns can vastly simplify the evaluation and transformation of trigonometric expressions.

Function Simplification

Function simplification is an essential skill in trigonometry and algebra. It involves expressing a complex function in a more simple, manageable form. For instance, simplifying the given function \( g(x) = 3 \csc x \) involves transforming \( \csc x \) into terms of sine: \( \csc x = \frac{1}{\sin x} \).
The step-by-step process of simplification looks as follows:

• Identify the trigonometric function like \( \csc x \).
• Use known identities to rewrite the function in terms of more commonly used functions like sine.
• Replace \( \csc x \) with \( \frac{1}{\sin x} \) in the expression \( g(x) = 3 \csc x \).
• Simplify the expression to \( g(x) = \frac{3}{\sin x} \).

Simplifying functions makes it easier to integrate, differentiate, or simply understand them within larger equations or systems. It's akin to decluttering an equation, making it more readable and easier to work with, thereby aiding comprehension and problem-solving.