Skip to main content
Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.7.106a

Deriving trigonometric identities
a. Differentiate both sides of the identity cos 2t = cos² t−sin² t to prove that sin 2 t= 2 sin t cos t.

Verified step by step guidance
1
Step 1: Differentiate the left side of the identity with respect to t. The left side is \( \cos(2t) \). Using the chain rule, the derivative is \(-2\sin(2t)\).
Step 2: Differentiate the right side of the identity with respect to t. The right side is \( \cos^2(t) - \sin^2(t) \). Use the chain rule and the power rule to differentiate each term separately.
Step 3: For \( \cos^2(t) \), use the chain rule: the derivative is \( 2\cos(t)(-\sin(t)) = -2\cos(t)\sin(t) \).
Step 4: For \( \sin^2(t) \), use the chain rule: the derivative is \( 2\sin(t)\cos(t) \).
Step 5: Combine the derivatives from steps 3 and 4: \(-2\cos(t)\sin(t) - 2\sin(t)\cos(t) = -2\sin(t)\cos(t) - 2\sin(t)\cos(t) = -4\sin(t)\cos(t)\). Set this equal to the derivative from step 1, \(-2\sin(2t)\), and simplify to show \( \sin(2t) = 2\sin(t)\cos(t) \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

Key Concepts

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

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. They serve as fundamental tools in calculus and can simplify complex expressions. Common identities include the Pythagorean identities, angle sum and difference identities, and double angle identities, which are essential for manipulating and proving relationships between sine and cosine functions.
Recommended video:
7:17
Verifying Trig Equations as Identities

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 applies specific rules, such as the derivatives of sine and cosine, to derive new relationships. This process is crucial for proving identities by showing that two expressions have the same derivative.
Recommended video:
05:53
Finding Differentials

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 sin(2t) = 2sin(t)cos(t) is derived from the sine and cosine functions. Understanding these formulas is essential for simplifying expressions and proving identities, as they provide a direct relationship between angles and their trigonometric values.
Recommended video:
Guided course
5:04
Converting between Degrees & Radians
Related Practice
Textbook Question

7–14. Find the derivative the following ways:

a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.

g(s) = 4s³ - 8s² +4s / 4s

224
views
Textbook Question

21–30. Derivatives

a. Use limits to find the derivative function f' for the following functions f.

f(s) = 4s³+3s; a= -3, -1

233
views
Textbook Question

79–82. {Use of Tech} Visualizing tangent and normal lines <IMAGE>

a. Determine an equation of the tangent line and the normal line at the given point (x0, y0) on the following curves. (See instructions for Exercises 73–78.)

x⁴ = 2x²+2y²; (x0, y0)=(2, 2) (kampyle of Eudoxus)

286
views
Textbook Question

Comparing velocities Two stones are thrown vertically upward, each with an initial velocity of 48 ft/s at time t=0. One stone is thrown from the edge of a bridge that is 32 feet above the ground, and the other stone is thrown from ground level. The height above the ground of the stone thrown from the bridge after t seconds is f(t) = − 16t²+48t+32. and the height of the stone thrown from the ground after t seconds is g(t) = −16t²+48t.

a. Show that the stones reach their high points at the same time.

304
views
Textbook Question

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = 1/x; P (1,1)

201
views
Textbook Question

Vertical tangent lines If a function f is continuous at a and lim x→a| f′(x)|=∞, then the curve y=f(x) has a vertical tangent line at a, and the equation of the tangent line is x=a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 71–72) is used. Use this information to answer the following questions.

73. {Use of Tech} Graph the following functions and determine the location of the vertical tangent lines.

a. f(x) = (x-2)^1/3

353
views