Exercise 56 Page 449

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a Notice that cos x and - 1 both equal y. Therefore, we can use the Substitution Method and equate the right-hand sides of the functions.

y= cos x and y= - 1 ⇓ cos x= - 1Let's solve for x in this equation.

cos x = - 1

cos^(-1)(LHS) = cos^(-1)(RHS)

x = cos^(- 1) - 1

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x = 180^(∘)

We can also solve this equation by graphing y =cos x and y=- 1 in the same coordinate plane and identifying where they intersect.

b If we extend the graph from Part A in the positive or negative horizontal direction, we can identify additional possible solutions.

In addition to the first solution at x=180^(∘), we have four more solutions at 540^(∘), 900^(∘), 1260^(∘), and 1620^(∘).

c If we examine the graph from Part B, wee see that each solution is one full cycle, or 360^(∘), apart.

Therefore, if we start with the first solution at x=180^(∘), to capture all solutions we have to add a factor of (360n)^(∘), where n is the number of cycles. x=180^(∘)+(360n)^(∘)

Subchapter links

9.1.1 Introduction to Periodic Models p.433-434

9.1.2 Graphing the Sine Function p.438-440

9.1.3 Unit Circle ↔ Graph p.442-444

9.1.4 Graphing and Interpreting the Cosine Function p.447-449

9.1.5 Defining a Radian p.452-453

9.1.6 Building a Unit Circle p.456-457

9.1.7 The Tangent Function p.460-461

Exercise 56 Page 449

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