교과서 질문
In Exercises 63–82, use a sketch to find the exact value of each expression. tan [cos⁻¹ (− 1/3)]
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Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
모든 교과서
Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions문제 75
Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions문제 752장, 문제 75
In Exercises 75–78, graph one period of each function. y = |2 cos x/2|
검증된 단계별 안내1
Identify the function to be graphed: \(y = |2 \cos \frac{x}{2}|\). Notice that the function involves the cosine of \(\frac{x}{2}\), scaled by 2, and then the absolute value is taken.
Determine the period of the inner cosine function. The standard cosine function \(\cos x\) has a period of \(2\pi\). Since the argument is \(\frac{x}{2}\), the period \(T\) is given by \(T = \frac{2\pi}{\frac{1}{2}} = 4\pi\).
Set the domain for one period of the function from \(x = 0\) to \(x = 4\pi\) to capture one full cycle of \(\cos \frac{x}{2}\).
Calculate key points within this interval where the cosine function reaches its maximum, minimum, and zeros. For example, find values of \(x\) where \(\cos \frac{x}{2} = 1, 0, -1\), then apply the transformations: multiply by 2 and take the absolute value.
Plot these points and sketch the graph by connecting them smoothly, remembering that the absolute value makes all negative values positive, so the graph will be reflected above the x-axis wherever \(2 \cos \frac{x}{2}\) is negative.
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주요 개념
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Period of a Trigonometric Function
The period of a trigonometric function is the length of the interval over which the function completes one full cycle before repeating. For functions like cosine, the standard period is 2π, but it changes when the function's argument is scaled, such as cos(bx), where the period becomes 2π/|b|.
추천 영상:
5:33
Period of Sine and Cosine Functions
Effect of Absolute Value on Trigonometric Graphs
Applying the absolute value to a trigonometric function, like y = |f(x)|, reflects all negative values of the function above the x-axis, making the entire graph non-negative. This transformation alters the shape by 'folding' the parts below the x-axis upward, affecting the graph's appearance but not its period.
추천 영상:
5:08Sine, Cosine, & Tangent of 30°, 45°, & 60°
Graphing Scaled Trigonometric Functions
When the input of a trigonometric function is scaled, such as cos(x/2), the graph stretches horizontally, increasing the period. Understanding how the coefficient inside the function affects the x-axis scaling is essential for accurately plotting one period of the function.
추천 영상:
6:04Introduction to Trigonometric Functions
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In Exercises 67–68, use the method of adding y-coordinates to graph each function for 0 ≤ x ≤ 4. y = cos πx + sin π/2 x
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In Exercises 63–82, use a sketch to find the exact value of each expression. _ sin (cos⁻¹ √2/2)
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In Exercises 63–82, use a sketch to find the exact value of each expression. cos [tan⁻¹ (− 2/3)]
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In Exercises 79–82, graph f, g, and h in the same rectangular coordinate system for 0 ≤ x ≤ 2π. Obtain the graph of h by adding or subtracting the corresponding y-coordinates on the graphs of f and g. f(x) = 2 cos x, g(x) = cos 2x, h(x) = (f + g)(x)
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교과서 질문
In Exercises 75–78, graph one period of each function. y = −|3 sin πx|
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