Trigonometry Assignment

This assignment focuses on various topics in Trigonometry including polar expressions, rectangular coordinates, polar points, vector components, and more.

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 °