Unformatted text preview: w1 = − 2+ i 2, w2 = − 2 − i 2,
to
√
and w3 = 2 − i 2. 854 Applications of Trigonometry √
√
3. For z = 2+ i 2, we have z = 2cis π . With r = 2, θ = π and n = 3 the usual computations
4
4
√
√
√
√
π
π
7π
yield w0 = 3 2cis 12 , w1 = 3 2cis 9π = 3 2cis 34 and w2 = 3 2cis 112 . If we were
12
to convert these to rectangular form, we would need to use either the Sum and Diﬀerence
Identities in Theorem 10.16 or the HalfAngle Identities in Theorem 10.19 to evaluate w0 and
w2 . Since we are not explicitly told to do so, we leave this as a good, but messy, exercise.
4. To ﬁnd the ﬁve ﬁfth roots of 1, we write 1 = 1cis(0). We have r = 1, θ = 0 and n = 5.
√
π
π
π
Since 5 1 = 1, the roots are w0 = cis(0) = 1, w1 = cis 25 , w2 = cis 45 , w3 = cis 65 and
π
w4 = cis 85 . The situation here is even graver than in the previous example, since we have
π
no identities developed to help us determine the cosine or sine of 25 . At this stage, we could
approximate our answers using a calculator, and we leave this to the Exercises.
Now that we have done some computations using Theorem 11.17, we take a step back to look
at things geometrically. Es...
View
Full Document
 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

Click to edit the document details