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

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Step 1: Observe the shape of the graphed function. Identify if it resembles the sine wave (which starts at zero and goes up) or the cosine wave (which starts at a maximum or minimum).

Step 2: Recall the basic properties of sine and cosine functions: sine is an odd function, meaning \(\sin(-x) = -\sin(x)\), and cosine is an even function, meaning \(\cos(-x) = \cos(x)\).

Step 3: Check the symmetry of the graph. If the graph is symmetric about the y-axis, the function is even. If it is symmetric about the origin, the function is odd.

Step 4: Match the graph's starting point and symmetry to the corresponding basic trigonometric function (sine or cosine) and determine its parity (even or odd).

Step 5: Conclude by stating the identified function and whether it is even or odd based on the observations and properties.

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Basic Trigonometric Functions

The primary trigonometric functions are sine, cosine, and tangent, each representing ratios of sides in a right triangle or coordinates on the unit circle. Recognizing their characteristic waveforms—sine and cosine as smooth periodic waves and tangent with vertical asymptotes—is essential for identifying graphed functions.

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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. Understanding these properties helps classify trigonometric functions: cosine is even, sine and tangent are odd.

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Even and Odd Identities

Graphical Characteristics of Trigonometric Functions

Each trig function has distinct graph features: sine starts at zero and oscillates between -1 and 1, cosine starts at 1, and tangent has repeating vertical asymptotes. Identifying these traits in a graph aids in determining the function type and its symmetry.

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Pratica correlata

Domanda del libro di testo

Verify that each equation is an identity.

(1 - cos θ)/(1 + cos θ) = 2 csc² θ - 2 csc θ cot θ - 1

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Domanda del libro di testo

Find the remaining five trigonometric functions of θ.