Written_Homework4-Freya_Vickoren
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School
Oregon State University, Corvallis *
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Course
112Z
Subject
Mathematics
Date
Apr 3, 2024
Type
Pages
4
Uploaded by BrigadierToadPerson1091
MATH 112Z
Written Homework 4
- Unit 4 - Vectors and Identities Fall 2023 NAME: SECTION: 1)
A wilderness explorer leaves from a mountain lodge for an overnight hike. They hike due north for 3 hours. Then they hike northwest for 1 hour and stop to have lunch. After lunch, they head in the direction 10 degrees east of south, and hike for 4 hours, at which time they pitch their tent and settle in for the night. The explorer always hikes at a steady rate of 3 miles per hour. a.) Write three vectors, in component form, to represent the three parts of the hike. b.) How far from the mountain lodge is the explorer at the end of the day? c.) In what direction should the hiker go the next morning to get back to the mountain lodge? 2)
Each set of three points forms a triangle. Write three vectors to represent the three sides of the triangle. Is the triangle a right triangle? Show work using dot products. a.) Point A: (-2, -1) Point B: (-2, 8) Point C: (8, -1) b.) Point D: (3, 4) Point E: (3, 12) Point F: (6, 5)
3)
Two sled dogs are pulling on the same sled, and the sled needs to travel due North. The first dog, Chuck, is pulling with 135 pounds of force at an angle 15 degrees West of North. The second dog, Rufus, is pulling at an angle of 40 degrees East of South. a.) Sketch a diagram of the situation. b.) Write the component form for Chuck's force vector. c.) How many pounds of force will Rufus need to pull with in order to keep the sled moving due North? d.) What is the magnitude of the net force acting on the sled?
4)
Use DESMOS to help you determine whether each equation is a contradiction, a conditional equation, or a possible identity. a.) !"#(%)
’(#)*(%)
=
’(#)*(%)
!"#(%)
b.) cos(2࠵?) = 1 − sin
5
(࠵?)
c.) csc
5
(࠵?)61 + sin
5
(࠵?)8 = cot
5
(࠵?)
5)
Verify the following identities. a.) !"#(%):#;!(%)
#;!(%)
= −sin
5
(࠵?)
b.) sec
5
(࠵?) − tan
5
(࠵?) = 1
c.) (csc(࠵?) + cot(࠵?))
5
=
(!"#(%)(’)
?
sin
?
(%)
d.) (#)*(%):!"#(%))
?
!"#(%)
= sec(−࠵?) + 2 sin(−࠵?)
¡
Contradiction ¡
Conditional Eq.
¡
Poss. Identity
¡
Contradiction ¡
Conditional Eq.
¡
Poss. Identity
¡
Contradiction ¡
Conditional Eq.
¡
Poss. Identity
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6)
In calculus, we often need to take an expression of the form √࠵?
5
+ ࠵?
5
, √࠵?
5
− ࠵?
5
, or √࠵?
5
− ࠵?
5
, where ࠵?
is a variable and ࠵?
is a constant, and re-write it as a trigonometric expression. We will do this by drawing a right triangle with ࠵?
and ࠵?
as appropriate sides, using the Pythagorean Theorem, and substituting trigonometric expressions for ࠵?
as follows: a.) Sketch a right triangle with an acute angle ࠵?
. Label the side opposite ࠵?
as ࠵?
, and the side adjacent to ࠵?
as having a length of 4 units. b.) Determine the length of the hypotenuse of the above triangle, and label it on your diagram. Note: This length will be represented by an algebraic expression, not a number. c.) Use the triangle to determine tan(࠵?)
. d.) Use the triangle to show that √࠵?
5
+ 16
= 4 sec(࠵?)
. e.) Use a similar process (with a new triangle) to show that √25 − ࠵?
5
= 5 cos(࠵?)
. f.) Use a similar process (with a new triangle) to write √E:%
?
%
as a function of ࠵?
.