Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.7.106c

Chapter 3, Problem 3.7.106c

Deriving trigonometric identities
c. Differentiate both sides of the identity sin 2t = 2 sin t cost to prove that cos 2t = cos²t−sin²t.

Verified step by step guidance

Start by differentiating both sides of the given identity with respect to t. The identity is sin(2t) = 2 sin(t) cos(t).

Differentiate the left side: The derivative of sin(2t) with respect to t is cos(2t) multiplied by the derivative of 2t, which is 2. So, the derivative is 2 cos(2t).

Differentiate the right side: Use the product rule for differentiation, which states that the derivative of u(t)v(t) is u'(t)v(t) + u(t)v'(t). Here, u(t) = 2 sin(t) and v(t) = cos(t).

Calculate the derivatives: The derivative of 2 sin(t) is 2 cos(t), and the derivative of cos(t) is -sin(t). Apply the product rule: 2 cos(t) * cos(t) + 2 sin(t) * (-sin(t)).

Simplify the expression: This results in 2 cos²(t) - 2 sin²(t). Equate this to the derivative of the left side, 2 cos(2t), to show that cos(2t) = cos²(t) - sin²(t).

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Key Concepts

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

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are fundamental in simplifying expressions and solving equations in trigonometry. Common identities include the Pythagorean identities, angle sum and difference identities, and double angle identities, which are essential for deriving and proving relationships between different trigonometric functions.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In the context of trigonometric functions, differentiation allows us to compute the slopes of sine and cosine functions, which is crucial for proving identities and understanding their behavior. The derivatives of sin(t) and cos(t) are cos(t) and -sin(t), respectively, and these rules are applied when differentiating both sides of the given identity.

Double Angle Formulas

Double angle formulas are specific trigonometric identities that express trigonometric functions of double angles in terms of single angles. For example, the formula cos(2t) can be expressed as cos²(t) - sin²(t) or in other forms like 2cos²(t) - 1 or 1 - 2sin²(t). These formulas are derived from the basic trigonometric identities and are useful in simplifying expressions and proving other identities, such as the one in the given question.

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Textbook Question

62–65. {Use of Tech} Graphing f and f'

c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.

f(x)=(x²−1)sin^−1 x on [−1,1]

Textbook Question

Derivatives using tables Let $h(x)=f(g(x))$ and $p(x)=g(f(x))$ . Use the table to compute the following derivatives.

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c. $p prime of 4$

Textbook Question

Throwing a stone Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 32 ft/s from a height of 48 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t²+32t+48.

c. What is the height of the stone at the highest point?

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Textbook Question

Vibrations of a spring Suppose an object of mass m is attached to the end of a spring hanging from the ceiling. The mass is at its equilibrium position $y equals 0$ when the mass hangs at rest. Suppose you push the mass to a position $y sub 0$ units above its equilibrium position and release it. As the mass oscillates up and down (neglecting any friction in the system), the position y of the mass after t seconds is $y = y_{0} \cos (t \sqrt{\frac{k}{m}})$ , where $k is greater than 0$ is a constant measuring the stiffness of the spring (the larger the value of $k$ , the stiffer the spring) and $y$ is positive in the upward direction.

Use equation (4) to answer the following questions.

c. How would the velocity be affected if the experiment were repeated with a spring having four times the stiffness ( $k$ is increased by a factor of $4$ )?

Textbook Question

Finding derivatives from a table Find the values of the following derivatives using the table. <IMAGE>

c. d/dx ((f(x)g(x)) |x=3

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Textbook Question

Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>

c. Solve the equation y(x²+4)=8 for y to find an explicit expression for y and then calculate dy/dx.

Deriving trigonometric identitiesc. Differentiate both sides of the identity sin 2t = 2 sin t cost to prove that cos 2t = cos²t−sin²t.

Verified video answer for a similar problem:

Key Concepts

Trigonometric Identities

Differentiation

Double Angle Formulas

Deriving trigonometric identities
c. Differentiate both sides of the identity sin 2t = 2 sin t cost to prove that cos 2t = cos²t−sin²t.