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Trigonometry Week 14 Test Grade Report

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Trigonometry

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Trigonometry Week 14 Test - Grade Report

Score:

79% (79.3333 of 100 pts)

Submitted:

Aug 8 at 12:28pm

Question: 1

Grade: 0.33333334 /

4.0

Find the cube roots of −4+43–√i−4+43i.

22(cos4π9+isin4π9)cos4π9+isin4π9, 22(cos2π9+isin2π9)cos2π9+isin2π9,

22(cos8π9+isin8π9)cos8π9+isin8π9 (33%)

Solution

Write −4+43–√i−4+43i in polar form

−4+43–√i−4+43i=88(cos2π3+isin2π3)cos2π3+isin2π3

To find the cube roots, use the formula

r√n(cos(θ+2πkn)+isin(θ+2πkn))rncosθ+2πkn+isinθ+2πknwhere k=0,1,2,3,...,n−1.k=0,1,

2,3,...,n−1.

z=z=22⎛⎝⎜cos⎛⎝⎜2π3+2πk3⎞⎠⎟+isin⎛⎝⎜2π3+2πk3⎞⎠⎟⎞⎠⎟cos2π3+2πk3+isin2

π3+2πk3where k=0,1,2.k=0,1,2.

When k=0k=0

22⎛⎝⎜cos⎛⎝⎜2π3+2π(0)3⎞⎠⎟+isin⎛⎝⎜2π3+2π(0)3⎞⎠⎟⎞⎠⎟cos2π3+2π03+isin2

π3+2π03=22(cos2π9+isin2π9)cos2π9+isin2π9

When k=1k=1

22⎛⎝⎜cos⎛⎝⎜2π3+2π(1)3⎞⎠⎟+isin⎛⎝⎜2π3+2π(1)3⎞⎠⎟⎞⎠⎟cos2π3+2π13+isin2

π3+2π13=22(cos8π9+isin8π9)cos8π9+isin8π9

When k=2k=2

22⎛⎝⎜cos⎛⎝⎜2π3+2π(2)3⎞⎠⎟+isin⎛⎝⎜2π3+2π(2)3⎞⎠⎟⎞⎠⎟cos2π3+2π23+isin2

π3+2π23=22(cos14π9+isin14π9)cos14π9+isin14π9

Therefore, the cube roots of −4+43–

√i−4+43i are 22(cos2π9+isin2π9)cos2π9+isin2π9, 22(cos8π9+isin8π9)cos8π9+isin8π9

, 22(cos14π9+isin14π9)cos14π9+isin14π9.

Question: 2

Grade: 1.0 / 4.0

Solve.

u4+7=32u4+7=32

±5–√,±i5–√±5,±i5 (100%)

Solution

Set the equation equal to zero and solve by factoring.

u4+7=32u4+7=32

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u4=25u4=25

u4−25=0u4−25=0

(u2+5)(u2−5)=0u2+5u2−5=0

u2+5=0u2+5=0 and u2−5=0u2−5=0

u2=−5u2=−5and u2=5u2=5

u=±i5–√u=±i5 and u=±5–√u=±5

Therefore, the four solutions are ±5–√,±i5–√±5,±i5.

Question: 3

Grade: 1.0 / 4.0

Find the quotient z1z2z1z2.

z1=42(cos180°+isin180°)z1=42cos180°+isin180°and z2=6(cos90°+isin90°)z2=6cos90°

+isin90°

z1z2=7(cos90°+isin90°)z1z2=7cos90°+isin90° (100%)

Solution

z1z2z1z2

r1r2(cos(θ1−θ2)+isin(θ1−θ2))r1r2cosθ1−θ2+isinθ1−θ2

426(cos(180°−90°)+isin(180°−90°))426cos180°−90°+isin180°−90°

7(cos90°+isin90°)7cos90°+isin90°

Question:

Grade: 1.0 / 4.0

Solve.

t4−6,561=0t4−6,561=0

−9−9, 99, 9i9i, −9i−9i (100%)

Solution

t4−6,561t4−6,561

t4t4

6,5616,561

Find r and θ.

In the complex plane, 6,561 is the point (6,561, 0) on the positive real axis, so r = 6,561

and θ = 2π;.

Use

r√n(cos(θ+2πkn)+isin(θ+2πkn))rncosθ+2πkn+isinθ+2πknwhere k=0,1,2,3,...,n−1.k=0,1,

2,3,...,n−1.

z=6,561−−−−√4(cos(2π+2πk4)+isin(2π+2πk4))z=6,5614cos2π+2πk4+isin2π+2πk4wher

e k=0,1,2,3.k=0,1,2,3.

When k=0:k=0:

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Trigonometry Week 14 Test - Grade Report Score: 79% (79.3333 of 100 pts) Submitted: Aug 8 at 12:28pm Question: 1 Grade: 0.33333334 / 4.0 Find the cube roots of −4+43–√i−4+43i. 22(cos4π9+isin4π9)cos4π9+isin4π9, 22(cos2π9+isin2π9)cos2π9+isin2π9, 22(cos8π9+isin8π9)cos8π9+isin8π9 (33%) Solution Write −4+43–√i−4+43i in polar form −4+43–√i−4+43i=88(cos2π3+isin2π3)cos2π3+isin2π3 To find the cube roots, use the formula r√n(cos(θ+2πkn)+isin(θ+2πkn))rncosθ+2πkn+isinθ+2πknwhere k=0,1,2,3,...,n−1.k=0,1, 2,3,...,n−1. z=z=22⎛⎝⎜cos⎛⎝⎜2π3+2πk3⎞⎠⎟+isin⎛⎝⎜2π3+2πk3⎞⎠⎟⎞⎠⎟cos2π3+2πk3+isin2 π3+2πk3where k=0,1,2.k=0,1,2. When k=0k=0 22⎛⎝⎜cos⎛⎝⎜2π3+2π(0)3⎞⎠⎟+isin⎛⎝⎜2π3+2π(0)3⎞⎠⎟⎞⎠⎟cos2π3+2π03+isin2 π3+2π03=22(cos2π9+isin2π9)cos2π9+isin2π9 When k=1k=1 22⎛⎝⎜cos⎛⎝⎜2π3+2π(1)3⎞⎠⎟+isin⎛⎝⎜2π3+2π(1)3⎞⎠⎟⎞⎠⎟cos2π3+2π13+isin2 π3+2π13=22(cos8π9+isin8π9)cos8π9+isin8π9 When k=2k=2 22⎛⎝⎜cos⎛⎝⎜2π3+2π(2)3⎞⎠⎟+isin⎛⎝⎜2π3+2π(2)3⎞⎠⎟⎞⎠⎟cos2π3+2π23+isin2 π3+2π23=22(cos14π9+isin14π9)cos14π9+isin14π9 Therefore, the cube roots of −4+43– √i−4+43i are 22(cos2π9+isin2π9)cos2π9+isin2π9, 22(cos8π9+isin8π9)cos8π9+isin8π9 , 22(cos14π9+isin14π9)cos14π9+isin14π9. Question: 2 Grade: 1.0 / 4.0 Solve. u4+7=32u4+7=32 ±5–√,±i5–√±5,±i5 (100%) Solution Set the ...
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