Order Analytic Trigonometry Essay

Order Analytic Trigonometry Essay

In working with identities, you generally start with the side of the identity

that looks more complicated. Then use your knowledge of well-known identities

and your algebra skills until that side looks exactly like the other side. Note: For

convenience LS and RS will be used to denote the left side and right side of an

identity, respectively.

434.

435.

436.

437.

438.

439.

440.

441.

442.

443.

444.

445.

446.

447.

448.

449.

450.

451.

452.

453.

454.

455.

456.

457.

458.

459. To verify that the statement is not an identity, show a counterexample. Let

= 0º. Then cos(2 ) = cos(2 · 0º) = cos(0º) = 1, but 2cos = 2cos0º = 2(1) =

2≠1. Thus, cos(2 ) = 2cos is not an identity.

460. Let = 30º and β = 60º. Then sin( β) = sin(30º 60º) = sin 90º = 1, but

. Thus sin( β) = sin sin β is not an

identity.

461. Yes, it is a solution.

462.

Thus, it is not a solution.

463. Solve for sin x,

Now, determine all values for x in [0, 2π) for which sin x = . The reference

angle is .

The sine function is positive in quadrants I and II. Thus,

464.

Now, determine all values for x in (–∞, ∞) for which tan x = –1. The reference

angle is . The tangent function is negative in quadrants II and IV. Thus, and

are the only two values in [0, 2π) for which tan x = –1. Because the tangent

function has a period of π, all solutions to sin x cos x = 0 can be expressed as

and , where n is an integer. Note: Because and

, division by cos x in this problem was permissible.

465.

Now, determine all values for x in [0, 2π) for which either sin x = 1 or tan x = 1.

For sin x = 0, the reference angle is 0, so 0 and π are the only two values in [0,

2π) for which sin x = 0.

For tan x = 1, the reference angle is . The tangent function is positive in

quadrants I and III, so and are the only two values in [0, 2π) for which tan x

= 1. Therefore, sin x tan x = sin x has solutions

466.

Now, determine all values for x in [0, 2π) for which either or .

The reference angle is . The solutions are .

467.

Now, determine all values for x in [0, 2π) for which cos x = 0. The reference

angle is , so the solutions are

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