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Ch. 5 - Trigonometric Identities
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 5 - Trigonometric IdentitiesProblem 5.1.30
Chapter 6, Problem 5.1.30

Identify the basic trigonometric function graphed, and determine whether it is even or odd.


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Verified step by step guidance
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Step 1: Observe the shape and key features of the graphed function, such as its period, amplitude, and where it crosses the axes. This will help identify which basic trigonometric function it resembles (sine, cosine, tangent, etc.).
Step 2: Recall the basic graphs of sine and cosine functions: sine starts at zero and goes up, cosine starts at a maximum value. Tangent has vertical asymptotes and repeats every \( \pi \). Compare these characteristics to the graph you have.
Step 3: Once you identify the function (for example, sine or cosine), use the definition of even and odd functions to determine its symmetry. A function \( f(x) \) is even if \( f(-x) = f(x) \) and odd if \( f(-x) = -f(x) \).
Step 4: Recall that the sine function is an odd function, meaning its graph is symmetric about the origin, while the cosine function is even, symmetric about the y-axis. Tangent is also an odd function.
Step 5: Use the symmetry observed in the graph to conclude whether the function is even or odd based on the above definitions and your identification of the function.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Basic Trigonometric Functions

The primary trigonometric functions are sine, cosine, and tangent, each with distinct shapes and properties. Recognizing the graph involves identifying characteristic features like amplitude, period, and intercepts. For example, sine and cosine functions have smooth, periodic waves, while tangent has vertical asymptotes.
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Introduction to Trigonometric Functions

Even and Odd Functions

A function is even if f(-x) = f(x), meaning its graph is symmetric about the y-axis. It is odd if f(-x) = -f(x), showing symmetry about the origin. Determining this helps classify the trigonometric function, as cosine is even and sine is odd.
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Even and Odd Identities

Graphical Analysis of Trigonometric Functions

Analyzing the graph involves observing symmetry, periodicity, and key points like zeros and maxima. This helps identify the function type and its properties. For instance, cosine graphs peak at x=0, while sine graphs cross the origin, aiding in function identification.
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Introduction to Trigonometric Functions
Related Practice
Textbook Question

Find the remaining five trigonometric functions of θ.

cos θ = 1/5, θ in quadrant I

Textbook Question

Verify that each equation is an identity.

(1 + sin θ)/(1 - sin θ) - (1 - sin θ)/( 1 + sin θ) = 4 tan θ sec θ

Textbook Question

Factor each trigonometric expression.

cot⁴ x + 3 cot² x + 2

Textbook Question

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

cos θ (cos θ - sec θ)

Textbook Question

Verify that each equation is an identity.

1/(sec α - tan α) = sec α + tan α

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Textbook Question

Simplify each expression.

± √[(1 + cos (x/4))/2]