Pre-Calc Exam Notes 66

# Pre-Calc Exam Notes 66 - r 2(which we can do since r> 0...

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66 Chapter 3 Identities §3.1 Note how we proved the identity by expanding one of its sides ( sin θ cos θ ) until we got an expres- sion that was equal to the other side (tan θ ). This is probably the most common technique for proving identities. Taking reciprocals in the above identity gives: cot θ = cos θ sin θ when sin θ n= 0 (3.2) x y 0 θ | y | | x | r ( x , y ) Figure 3.1.1 We will now derive one of the most important trigonometric identities. Let θ be any angle with a point ( x , y ) on its terminal side a distance r > 0 from the origin. By the Pythagorean The- orem, r 2 = x 2 + y 2 (and hence r = r x 2 + y 2 ). For example, if θ is in QIII as in Figure 3.1.1, then the legs of the right triangle formed by the reference angle have lengths | x | and | y | (we use absolute values because x and y are negative in QIII). The same argument holds if θ is in the other quadrants or on either axis. Thus, r 2 = | x | 2 + | y | 2 = x 2 + y 2 , so dividing both sides of the equation by
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Unformatted text preview: r 2 (which we can do since r > 0) gives r 2 r 2 = x 2 + y 2 r 2 = x 2 r 2 + y 2 r 2 = p x r P 2 + p y r P 2 . Since r 2 r 2 = 1, x r = cos θ , and y r = sin θ , we can rewrite this as: cos 2 θ + sin 2 θ = 1 (3.3) You can think of this as sort of a trigonometric variant of the Pythagorean Theorem. Note that we use the notation sin 2 θ to mean (sin θ ) 2 , likewise for cosine and the other trigono-metric functions. We will use the same notation for other powers besides 2. From the above identity we can derive more identities. For example: sin 2 θ = 1 − cos 2 θ (3.4) cos 2 θ = 1 − sin 2 θ (3.5) from which we get (after taking square roots): sin θ = ± r 1 − cos 2 θ (3.6) cos θ = ± r 1 − sin 2 θ (3.7)...
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## This note was uploaded on 01/21/2012 for the course MAC 1130 taught by Professor Dr.cheun during the Fall '11 term at FSU.

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