Ch. 1 - Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 1 - Functions

Problema 1.3.50

Capitolo 1, Problema 1.3.50

Using the Half-Angle Formulas

Find the function values in Exercises 47–50.

sin² 3π/8

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First, recognize that the half-angle formulas are useful for finding the sine or cosine of an angle that is half of a given angle. The half-angle formula for sine is: \( \sin^2 \theta = \frac{1 - \cos(2\theta)}{2} \).

In this problem, we need to find \( \sin^2 \frac{3\pi}{8} \). Notice that \( \frac{3\pi}{8} \) is half of \( \frac{3\pi}{4} \). Therefore, we can use the half-angle formula for sine.

Substitute \( \theta = \frac{3\pi}{8} \) into the half-angle formula: \( \sin^2 \frac{3\pi}{8} = \frac{1 - \cos(\frac{3\pi}{4})}{2} \).

Next, find \( \cos(\frac{3\pi}{4}) \). The angle \( \frac{3\pi}{4} \) is in the second quadrant where cosine is negative. The reference angle is \( \frac{\pi}{4} \), and \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \). Therefore, \( \cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2} \).

Substitute \( \cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2} \) back into the formula: \( \sin^2 \frac{3\pi}{8} = \frac{1 - (-\frac{\sqrt{2}}{2})}{2} \). Simplify the expression to find the value of \( \sin^2 \frac{3\pi}{8} \).

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Half-Angle Formulas

Half-angle formulas are trigonometric identities that express the sine and cosine of half an angle in terms of the sine and cosine of the original angle. For example, the sine half-angle formula is given by sin(θ/2) = √((1 - cos(θ))/2). These formulas are particularly useful for simplifying expressions involving angles that are not easily computable.

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Converting between Degrees & Radians

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. They are fundamental in calculus and are used to model periodic phenomena. Understanding how to evaluate these functions at specific angles, including those expressed in radians, is essential for solving problems involving trigonometric identities.

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Introduction to Trigonometric Functions

Radians and Degrees

Radians and degrees are two units for measuring angles. Radians are based on the radius of a circle, where one radian is the angle subtended by an arc equal in length to the radius. Understanding the conversion between these units is crucial, especially when working with trigonometric functions, as many calculus problems use radians for angle measurements.

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Pratica correlata

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[Technology Exercise]

a. Graph the functions f(x) = x/2 and g(x) = 1 + (4/x) together to identify the values of x for which

x/2 > 1 + 4/x

b. Confirm your findings in part (a) algebraically.

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Finding a Viewing Window

In Exercises 5–30, find an appropriate graphing software viewing window for the given function and use it to display that function’s graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function.

y = x + (1/10) sin 30x

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Shifting Graphs

The accompanying figure shows the graph of y = −x² shifted to two new positions. Write equations for the new graphs.

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Refer to the given figure. Write the radius r of the circle in terms of α and θ.

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