Ch. 1 - Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 1 - Functions

Problem 58a

Chapter 1, Problem 58a

Apply the formula for cos (A − B) to the identity sin θ = cos (π/2 − θ) to obtain the addition formula for sin (A + B).

Verified step by step guidance

Start by recalling the identity for cosine of a difference: \( \cos(A - B) = \cos A \cos B + \sin A \sin B \).

Apply the identity \( \sin \theta = \cos(\frac{\pi}{2} - \theta) \) to the formula for \( \cos(A - B) \). Substitute \( A = \frac{\pi}{2} \) and \( B = \theta \) into the formula.

This substitution gives \( \cos(\frac{\pi}{2} - \theta) = \cos \frac{\pi}{2} \cos \theta + \sin \frac{\pi}{2} \sin \theta \).

Since \( \cos \frac{\pi}{2} = 0 \) and \( \sin \frac{\pi}{2} = 1 \), simplify the expression to \( \cos(\frac{\pi}{2} - \theta) = 0 \cdot \cos \theta + 1 \cdot \sin \theta = \sin \theta \).

Now, to find the addition formula for \( \sin(A + B) \), use the identity \( \sin \theta = \cos(\frac{\pi}{2} - \theta) \) and apply it to \( \sin(A + B) = \cos(\frac{\pi}{2} - (A + B)) \). Substitute \( A + B \) into the formula for \( \cos(A - B) \) to derive \( \sin(A + B) = \sin A \cos B + \cos A \sin B \).

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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. The identity sin θ = cos (π/2 − θ) is a co-function identity that relates sine and cosine, illustrating how these functions are interconnected.

Cosine of a Difference Formula

The cosine of a difference formula states that cos(A - B) = cosA cosB + sinA sinB. This formula is essential for deriving other trigonometric identities and is used to express the cosine of the difference of two angles in terms of the sines and cosines of those angles. It serves as a foundational tool in proving various trigonometric relationships.

Sine Addition Formula

The sine addition formula states that sin(A + B) = sinA cosB + cosA sinB. This formula allows us to express the sine of the sum of two angles in terms of the sines and cosines of the individual angles. It is crucial for solving problems involving the addition of angles and is derived using the cosine of a difference formula and other trigonometric identities.

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Textbook Question

Derive a formula for tan (A − B).

Textbook Question

A triangle has side c = 2 and angles A = π/4 and B = π/3. Find the length a of the side opposite A.

Textbook Question

The law of sines The law of sines says that if a, b, and c are the sides opposite the angles A, B, and C in a triangle, then

(sin A) / a = (sin B) / b = (sin C) / c

Use the accompanying figures and the identity sin (π − θ) = sin θ, if required, to derive the law.

Textbook Question

Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).

e. 𝔂 = ƒ( x ) - 4

Textbook Question

In Exercises 55–58, graph each function, not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.15–1.17, and applying an appropriate transformation.

y = - √(1 + x/2)

Textbook Question

In Exercises 59–62, sketch the graph of the given function. What is the period of the function?

𝔂 = cos πx/2