Trigonometry Assignment: Vectors, Polar and Rectangular Forms

This trigonometry assignment presents solutions to problems involving polar and rectangular coordinate conversions, vector operations, and polar functions. The assignment begins with converting a polar point to rectangular coordinates and finding alternative polar expressions. It then explores filling in blanks for a given polar function, graphing the function, and finding specific points. The assignment also covers converting rectangular points to polar coordinates, resolving vectors into components, finding vector sums, and calculating the angle a vector sum makes with the positive x-axis. Furthermore, it includes finding vector differences, magnitudes, dot products, and angles between vectors, followed by drawing vectors and calculating their magnitudes, sums, and angles between them. The solutions provide detailed calculations and graphical representations to aid understanding of the concepts.

Assignment
Trigonometry
<STUDENT NAME>
APRIL 15, 2019

Problem 1. SHOW ALL WORK. Given the polar point (4, -19π/6)
1A. Find two other polar expressions for the point, one with a negative r value, and another with a
positive r value. Use exact values.
4 ( cos ( −19 π
6 ) i+sin ( −19 π
6 ) j )
4 (− √3
2 i + 1
2 j )
¿−4 ( √3
2 i− 1
2 j )= (−4 , 11 π
6 )
4 (− √3
2 i+ 1
2 j )= (4 , 5 π
6 )
1B. Convert the polar point (4, -19π/6) into rectangular coordinates (x,y). Compute the
coordinates exactly.
Polar point is given by r (icos θ+ j sin θ)
r =4
θ= π
6
4 ( − √ 3
2 i+ 1
2 j )=−3.464 i+2 j
( x , y ) =(−3.46,2)
Problem 2. Fill in the blanks in the listed polar points (r, θ) below, to satisfy the polar function
r = 1 – 6cos(2θ). Find all of these points exactly.
 Given θ = π, find r: ( ____, π)
Sol.
Given r=1−6 cos 2 θ
r =1−6 cos−1 2 π
r =1−6
r =−5
 Given θ = π/8, find r: ( ¿¿, π/8)
Sol.
Given r=1−6 cos 2 θ

r =1−6 cos−1 2 π
8
r =1−6 cos π
4
r =1−6( 1
√ 2 )
r =1−3 √2=−3.24
 Find one specific point. Given r = 4, find θ: ( 4,____)
Sol.
Given r=1−6 cos 2 θ
4=1−6 cos 2 θ
cos 2 θ=−1
2
2 θ=cos−1 −1
2
2 θ= 2 π
3
θ= π
3
 Graph the polar function r = 1 – 6cos(2θ). Locate and label on your graph the three points
you found above in problem 2A.
Sol.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
Problem 3. Convert the rectangular point (-12, 0) into polar coordinates in the form (r, θ). Find exact
values.
Given ( x , y ) = ( −12,0 )
r = √ x2 + y2
r = √ 122 +02=12
tanθ= y
x =¿ θ=tan−1 y
x
tan−1 0
12 =π
(-5,π)
(-3.24,π/4)
(-4,π/3)

Therefore, polar coordinate is ( 12 , π ¿
Problem 4:
 4A. Vector W has a magnitude of 20 and when drawn in standard position, makes an angle
of 106° with the positive x axis. Draw vector W and resolve it into its component form.
Round your final answers to the nearest hundredth.
Ans)
w=w cos 106 ° i+w sin 106 j
w=−5.512 i+ 19.225 j
 4B. Vector Z has a magnitude of 14 and when drawn in standard position, makes an angle
of 320° with the positive x axis. Draw vector Z and resolve it into its component form.
Round your final answers to the nearest hundredth.
Ans)
z=z cos 320 ° i+ z sin320 j
z=10.724 i+−8.999 j
 4C. Draw the vector sum W + Z in standard position and find the component form Z of W
Ans)
w=−5.512 i+ 19.225 j
z=10.724 i+−8.999 j

z +w= ( 10.724 i+−8.999 j ) +(−5.512i+19.225 j)
z +w=5.212i+10.226 j
 4D. Find the angle the vector sum W + Z makes with the positive x axis. Round your final
answer to the nearest tenth of a degree.
Ans)
tan θ= 10.226
5.212 =1.962=¿ θ=63 °
Problem 5. Vector u is < 8, -5>. Vector v is < -3, 15 >
 5A. Find the vector v − u . Find exact values.
Ans) v=8 i−5 j
u=−3 i=15 j
v – u = -11i + 20j
 5B. Find the magnitude of vector v − u : || v − u ||. Find an exact value.
Ans)
r = √−112+202=22.825
 5C. Find the dot product u ∙ v . Find an exact value.
Ans)
v . u= ( 8i −5 j ) . (−3 i+15 j )
¿ ( 8 ×−3 ) + (−5 × 15 )
¿−24−75
¿−99

 5D. Find the angle θ between u and v . θ should be an angle between 0° and 180°, rounded to
the nearest hundredth of a degree.
Ans)
cos θ=⃗ u .⃗ v⃗
|u|⃗|v|
cos θ= −99
√89 √234
¿>θ=133.32 °
Problem 6. Pick a set of vectors u and v and do the following 5 items for
u = < 12, 2 > and v = < 2, -8>
 Draw each vector in standard position.
Ans)

 B. Find the magnitude of vector u: ||u||
Ans)
r = √ 122 +22=12.165
 C. Find the sum of vectors < u + v >, and sketch the graph of < u + v > in standard
position.
Ans)
u+ v=10 i−6 j
 D. Find the angle θ, in degrees, between the vectors u and v. θ should be between 0°
and 180°.
Ans)
v . u= ( 12 i+2 j ) . ( 2 i−8 j ) = ( 12× 2 ) + ( 2×−8 ) =24−16=8
cos θ=⃗ u .⃗ v⃗
|u|⃗|v|
cos θ= 8
√148 √68 =¿ θ=85.426 °