Step by step calculations of the weekly tutorial worksheets. Using exclusively these, my notes of theme 1-5, and the textbook questions which are all available on my profile, I attained a mark of 94% for SWK122 without watching a single Jeff Hanson video.
This tutorial covers pre-knowledge, in particular the sine rule, cosine rule, and solving of
simultaneous equations.
It is your responsibility to revise the topics where necessary.
Question 1
Consider a triangle with side lengths and angles as shown.
Determine the values of the unknown side lenghts and an-
α gles in the accompanying triangle in the case that
b
c (a) b = 40 mm and c = 15 mm.
(b) b = 80 mm and c = 50 mm.
30◦ (c) a = 80 mm and b = 50 mm.
β
a Draw appropriate sketches of the triangles to substantiate
your answers.
For any set of three parameters given, one of four situations prevails:
• No solution exists. This means that it is not possible to construct a triangle with the
given values.
• A unique solutions exists. This relates to the fact that exactly one triangle can be
constructed with the given values.
• There are two solutions. In this case there are two possible triangles that relate to the
given values. The two triangles also ”fit into each other”, like a puzzle.
• There are infinitely many solutions. This means that there are infinitely many triangles
that can be constructed with the given values. (This is the case when the three angles
are given and all triangles will be similar.)
,Question 2
The general form of a system of two linear equations in two unknowns x and y is given by
ax + by = c
px + qy = r
Consider the following three systems of linear equations
(a) x + y = 1 (b) x + y = 1 (c) x + y = 1
2x − 2y = 2 2x + 2y = 2 2x + 2y = 1
and answer the questions.
2.1 Solve all three systems of linear equations. Use at least 4 significant digits when rounding
values in intermediate steps and give your final answers to exactly 4 significant digits.
Recall that for any system of linear equations (regardless of the number of equations
or the number of unknowns), one of the following three situations hold:
• There is no solution.
• There is a unique (one) solution.
• There is a parametric solution (i.e. there are infinitely many solutions). This
means that one or more of the variables are arbitrary and the remaining variables
are expressed in terms of the arbitrary variables.
2.2 Check your answers in Question 2.1 by substituting your values into the given system
of equations and see if they satisfy both equations.
2.3 Interpret your solutions in terms of the graphs that the systems of equations represent.
Note that each equation in the given system represents a line in two-dimensional space.
Depending on the slopes of the lines, we find that the system of equations represents
two lines that
• intersect in one point (the slopes of the lines are different),
• intersect in infinitely many points (the slopes of the lines are equal and the lines
lie on each other) or
• does not intersect at all (the slopes of the lines are equal and the lines do not lie
on each other). In this case the lines are parallel.
Question 3
Solve, if possible, the following systems of linear equations.
(a) x + y + z = 1 (b) x + y = 1 (c) x + y + z = 1
2x − 2y + z = 2 2x + 2y = 2 2x − 2y + z = 2
x − 3y = 0 3x + 2y = 3 3x − y + 2z = 3
, Question I
a
is
stiso sins
4051130
β sinβ
a
β no solution
b
α
so
80 h3o hβ
s
30
β
a
β 53.1 or β 126.9
53.1 30 180 Or
96.9 23.1
8830 96.9 or 8 30 sina.li
9 9928mm 9 39.23MM
c α C 50 8012 21501180720530
so
44.40 MM
β go
44h48
I β
β 34.3 Or β 145.7
448 Ya
or
115.7
64.3 2
, 29 oct 4
2x 24 2
2 24 2
2x 24 2
42 4
2 1.000
4 0.000
b 2x 24 2
2
24 2
0
no solution
C 22 24 2
2 24 1
no solution
34 34
34 4 2 1
by 2492 2
34 34 0 1
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