Download Section 3.4: Solving Problems Using Acute Triangles

Math 20
Foundations
Chapter 3:
Acute Triangle
Trigonometry
&
Chapter 4:
Oblique Triangle
Trigonometry
Outcome FM20.5

Demonstrate understanding of the cosine law and sine law (including the ambiguous
case).
Indicators:
a. Identify and describe situations relevant to self, family, or community that involve triangles
without a right angle.

4.1, 3.2
b. Develop, generalize, explain, and apply strategies for determining angles or side lengths of
triangles without a right angle.

3.1, 4.1
c. Draw diagrams to represent situations in which the cosine law or sine law could be used to
solve a question.

3.2, 3.3, 3.4, 4.2
d. Explain the steps in a given proof of the sine law or cosine law.

3.1, 3.2, 3.3
e. Illustrate and explain how one, two, or no triangles could be possible for a given set of
measurements for two side lengths and the non-included angle in a proposed triangle.

4.3
f. Develop, generalize, explain, and apply strategies for determining the number of solutions
possible to a situation involving the ambiguous case.

4.3
g. Solve situational questions involving triangles without a right angle.

3.2, 3.3, 3.4
Chapter 3 & 4 Definitions
______________________: In any acute triangle,
______________________: In any acute triangle, a2 = b2 + c2 - 2bc cos A
b2 = a2 + c2 - 2ac cos B
c2 = a2 + b2 - 2ab cos C
______________________: A triangle that does not contain a 90° angle.
__________________ case of the sine law: A situation in which two triangles can be drawn,
given the available information; the ambiguous case may occur when the given measurements are the
lengths of two sides and the measure of an angle that is not contained by the two sides (SSA).
Trigonometry Review
Draw a right triangle and label the following parts:
 Right angle
 Short leg
 Long leg
 Hypotenuse
Label the triangle above ∆𝐴𝐵𝐶, label the vertices using capital letters and label the sides using lower
case letters. The right angle should be called 𝐶, and the hypotenuse should be called 𝑐.

When you know the lengths of two sides of a right triangle, you can find the length of the third
side using the Pythagorean Theorem:
𝑎2 + 𝑏 2 = 𝑐 2 where ‘𝑐’ is the length of the hypotenuse
Example 1 – Find the lengths of the indicated sides.
(a)
Review Trig Functions:
SOH (sin=opp/hyp) CAH (cos=adj/hyp) TOA(tan=opp/adj)
Label the opposite side, adjacent side and hypotenuse with respect to angle 𝛼.
𝛼
Example 2- Determine the value of each trigonometric ratio to four decimal places.
a) sin 33o
b) cos 27o
Example 3- Determine the measure of A to the nearest degree.
a) sin A = 0.8660
b) cos A = 0.8660
Example 4- Determine the value of x in each proportion.
𝒙
𝟏𝟓
𝟏𝟐
𝟏𝟔
a) =
b)
=
𝟖
𝟔
𝒙
𝟏𝟐
Calculating an unknown side given an angle and one known side:
Example 5- Find the value of 𝑥.
Calculating an unknown angle given two known sides:
Example 6- Find the value of 𝑥.
To “solve” a triangle means to find all of the unknown sides and angles.
Example 7- Solve the following triangle.
Chapter 3: Acute Triangle Trigonometry
Section 3.1: Exploring Side–Angle Relationships in Acute
Triangles
You have used the primary trigonometric ratios to determine side lengths and angle measures in right
triangles.
Recall: SOH CAH TOA

In the figure to the right the ______________________ are represented by lower case
letters and the ______________________ are represented by upper case letters

The hypotenuse is the side ____________________ from the right angle in a triangle.
Opposite means the side ___________________ from the angle we are using. Adjacent
means the side ___________________ the angle we are using (but not the hypotenuse).
SOH:
CAH:
TOA:
Can you use primary trigonometric ratios to determine unknown sides and angles in all acute
triangles?
The following diagram represents a general acute triangle.
A.
What are two equivalent expressions that represent the height of ∆𝑨𝑩𝑪?
B.
If you drew the height of ∆𝑨𝑩𝑪 from a different vertex, how would the expressions for
that height be different?

C.
Create an equation using the expression you created in part A. Show how your equation
can be written as a ratio so that each ratio in the equation involves a side and an angle.
 to write this as a ratio, we can divide both sides by sin B
 only one side is written as a ratio, but if we divide both sides by sin
C both sides will be fractions
D.
Show how you could determine the measure of ∠𝑭 in this acute triangle.
**Make sure your calculator is set on DEG (degrees)**
Key Ideas:

The ratios of
length of opposite side
sin(angle)
are ___________________ for all three side–angle
pairs in an acute triangle.

In an acute triangle, ∆ ABC,
3.1 Assignment: Nelson Foundations of Mathematics 11, Sec 3.1, pg. 117
Questions: 2, 4, workbook example 3a
Section 3.2: Proving and Applying the Sine Law
In Lesson 3.1, you discovered a side–angle relationship in acute triangles. Before this relationship can
be used to solve problems, it must be proven to work in all acute triangles. Consider Ben’s proof:
Why did Ben draw the
height AD?
Why did Ben create 2
different expressions
that involved AD?
Why is Ben able to set these
two expressions as equal?
What did Ben do to
rewrite this expression?
In steps 4 and 5, Ben drew a
different height BE and repeated
steps 2 and 3 for the right triangles
this created. Explain why.
Explain why he was able to equate
all three ratios in step 6 to create
the sine law.
Example 1 – A triangle has angles measuring 80o and 55o. The side opposite the 80o angle is 12.0 m in
length. Determine the length of the side opposite the 55o angle to the nearest tenth of a meter.
Example 2 – Toby uses chains attached to hooks on the ceiling and a winch to lift engines at his
father’s garage. The chains, the winch, and the ceiling are arranged as shown. Toby solved the
triangle using the sine law to determine the angle that each chain makes with the ceiling to the nearest
degree. He claims that 𝜃 = 40o and 𝛼 = 54o . Is he correct? Explain, and make any necessary
corrections.
Key Ideas:
 The sine law can be used to determine unknown side lengths or angle measures in acute
triangles.
 You can use the sine law to solve a problem modeled by an acute triangle when you know:
o ________ sides and the angle ___________________ a known side.
o two ___________________ and any ___________________.
 If you know the measures of two angles in a triangle, you can determine the third angle because
the angles must add to ________.

When determining ___________________, it is more convenient to use:

When determining ___________________, it is more convenient to use:
3.2 Assignment: Nelson Foundations of Mathematics 11, Sec 3.2, pg. 124-127

Questions: 3abc, 4, 5, 6ac, 8a, 15
Section 3.3 Proving and Applying the Cosine Law
The sine law cannot always help you determine unknown angle measures or side lengths. Consider
these triangles:
There are two unknowns in each pair of equivalent ratios, so the pairs cannot be used to solve for the
unknowns. Another relationship is needed. This relationship is called the ___________________
law, and it is derived from the Pythagorean Theorem.
Before this relationship can be used to solve problems, it must be proven to work in all acute triangles:
1. Start by drawing an acute triangle ∆ABC. Draw the height from A to BC and label the
intersection D. We will label BD=x and CD = y.
o Why did we draw the height?
2. Write two different expressions for height (or h2) using the
Pythagorean Theorem.
3.
4. Set the expressions to equal one another and solve for c2
o Why can we set the two expressions to equal one another?
5. Rewrite the expression using only variables a, b, c and y and simplify:

Why did we eliminate the variable x?
6. We must replace y. To do this we need to create an equivalent expression for y using only
variables from ∆ABC (try using a trig function):
𝑎𝑑𝑗
𝑐𝑜𝑠 =
ℎ𝑦𝑝
7. Substitute the expression you just developed in your previous equation:
Cosine Law: c2 = a2 + b2 - 2ab cos C
This form of the cosine law is useful because when you are given information about an acute triangle,
you usually get information about the length of its sides or the size of its angles. This form we have
developed includes only variables related to the sides and one angle of the triangle.
Let’s take another look at the triangles we examined at the beginning of the lesson.
We were not able to use the sine law to determine the missing parts of the triangles, but now that we
have learned the cosine law we may be able solve them.
 Show how you can use the cosine law to determine the unknown side 𝑞 in ∆𝑄𝑅𝑆.

Show how you can use the cosine law to determine the unknown ∠𝐹 in ∆𝐷𝐸𝐹.
Example 1 – Determine the length of 𝐶𝐵 to the nearest metre.
Example 2 – The diagram at the right shows the plan for a
roof, with support beam 𝐷𝐸 parallel to 𝐴𝐵. The local
building code requires the angle formed at the peak of a
roof to fall within a range of 70o to 80o so that snow and ice
will not build up. Will this plan pass the local building code?
First we need to make sure that the units are consistent (all
feet)
Key Ideas

The Cosine law can be used to determine an unknown side length or angle measure in an
acute triangle. a2 =
b2 =
c2 =


You can use the cosine law to solve a problem that can be modeled by an acute triangle when
you know:
two sides and the
all three sides.
contained angle
o The contained angle is the angle _________________ two known sides.
When using the cosine law to determine an angle, you can:
o substitute the known values first, then solve for the unknown angle.
o rearrange the formula to solve for the cosine of the unknown angle, then
substitute and evaluate.
3.3 Assignment: Nelson Foundations of Mathematics 11, Sec 3.3, pg. 136-139

Questions: 1, 4, 5, 6ac, 7ac, 8
Section 3.4: Solving Problems Using Acute Triangles
Example 1 – Two security cameras in a museum must be adjusted to monitor a new display of fossils.
The cameras are mounted 6 m above the floor, directly across from each other on opposite walls. The
walls are 12 m apart. The fossils are displayed in cases made of wood and glass. The top of the display
is 1.5 m above the floor. The distance from the camera on the left to the centre of the top of the display
is 4.8 m. Both cameras must aim at the centre of the top of the display. Determine the angles of
___________________, to the nearest degree, for each camera.
1.
Draw a diagram. Be careful about where you put the display
case. Label the angles of depression using Ѳ and α.
2. Use a trig function and the info you have to determine the size of Ѳ.
3. Use one of the laws we have learned about to determine the length of BD (or a).
4. You should now be able to use a trig function and the info we have for the right triangle on
the right to determine the size of α.
Example 2 – The world’s tallest free-standing totem pole is located in Beacon Hill Park in Victoria,
British Columbia. While visiting the park, Manuel wanted to determine the height of the totem pole, so
he drew a sketch and made some measurements:
(i)
(ii)
(iii)
I walked along the shadow of the totem pole and counted 42 paces,
estimating that each pace was about 1 m.
I estimated that the angle of elevation of the Sun was about 40o.
I observed that the shadow ran uphill, and I estimated that the
angle the hill made with the horizontal was about 5o.
Example 3 – Brendan and Diana plan to climb the cliff at Dry Island Buffalo Jump, Alberta. They need
to know the height of the climb before they start. Brendan stands at point B, as shown in the diagram.
He determines that ∠𝐴𝐵𝐶, the angle of elevation to the top of the cliff, is 76o. Then he estimates ∠𝐶𝐵𝐷,
the angle between the base of the cliff, himself, and Diana, who is standing at point D. Diana estimates
∠𝐶𝐷𝐵, the angle between the base of the cliff, herself, and Brendan. Determine the height of the cliff to
the nearest metre.
Key Ideas:
 The sine law, the cosine law, the primary
trigonometric ratios, and the sum of angles in a triangle may all be useful when solving
problems that can be modeled using acute triangles.
 Drawing a clearly labeled diagram makes it easier to select a strategy for solving a problem.
 To decide whether you
need to use the sine
law or the cosine law,
consider the
information given
about the triangle and
the measurement to be
determined.
3.4 Assignment: Nelson
Foundations of Mathematics
11, Sec 3.4, pg. 147-150
Questions: 3, 4, 5, 7, 9, 13
Chapter 4: Oblique Triangle Trigonometry
Section 4.1: Exploring the Primary Trigonometric Ratios of
Obtuse Angles

Until now, w have only worked with acute angles. We have used the trig ratios to determine
the side lengths and angle measures in ___________________ triangles, and we have used
the sine and cosine laws to determine the side lengths and angle measures in acute
___________________ triangles.
Use a calculator to complete the table:
sin (180o –
𝜽
𝟏𝟖𝟎𝒐 − 𝜽
sin 𝜽
𝜽)
cos 𝜽
cos (180o –
𝜽)
tan 𝜽
tan (180o –
𝜽)
100o
110o
120o
130o
140o
150o
160o
170o
180o
What relationships do you observe when comparing the trig ratios for obtuse angles with the
trig ratios for the related supplementary acute angles?


The sine ratios for supplementary angles are equal.
The cosine and tangent ratios for supplementary angles are opposites.
Key Ideas

There are relationships between the value of a primary trigonometric ratio for an acute angle
and the value of the same primary trigonometric ratio for the supplement of the acute angle.

For any acute angle, 𝜃,
sin 𝜃 =
cos 𝜃 =
tan 𝜃 =
4.1 Assignment: Nelson Foundations of Mathematics 11, Sec 4.1, pg. 163: Questions: 1-4
Section 4.2: Proving and Applying the Sine and Cosine Laws for
Obtuse Triangles
We have already shown that the sine law works for acute triangles. Now we are going to try to prove
the sine law for obtuse triangles. Follow the steps of the investigation to prove that the sine law also
applies to obtuse triangles.
A.
Draw an obtuse triangle ABC with height AD.
B.
Write equations for sin (180o – ∠𝑨𝑩𝑪) and sin C using the two right triangles.
C.
Use the transitive property to make the two expressions for AD equal to each other, then
create a ratio.
D.
Draw a new height, h, from B to base b in the triangle.
E.
Write equations for sin A and sin C using the two right triangles.
F.
Use the transitive property to make the two expressions for h equal to each other.
Example 1 – In an obtuse triangle, ∠B measures 23.0o and its opposite side, b, has a length of 40.0 cm.
Side a is the longest side of the triangle, with a length of 65.0 cm. Determine the measure of ∠𝐴 to the
nearest tenth of a degree.
In lesson 3.3 we proved the cosine law for acute triangles. Follow the steps of the investigation to
prove that the cosine law also applies to obtuse triangles.
A.
Use ∆𝑨𝑩𝑪 as shown below.
B.
Extend the base of the triangle to D, creating two overlapping right triangles, ∆𝑪𝑩𝑫 and
∆𝑨𝑩𝑫, with height BD. Note on your diagram that two angles are formed at C, ∠𝑨𝑪𝑩 and
∠𝑫𝑪𝑩, such that ∠𝑫𝑪𝑩 = 180o – ∠𝑨𝑪𝑩.
C.
Let side CD be x. Use the Pythagorean Theorem to write two expressions for h2, using the
two right triangles.
D.
Use the transitive property to make the two expressions for h2 equal to each other. Rearrange to isolate c2.
E.
In the small right triangle, use a primary trig ratio to write an expression for x.
F.
Substitute your expression for x into your equation from part D.
Remember that cos (180° - ACB) = -cos ACB:
Example 3 – The roof of a house consists of two slanted sections as
shown. A roofing cap is being made to fit the crown of the roof,
where the two slanted sections meet. Determine the measure
of the angle needed for the roofing cap, to the nearest tenth of a
degree.
Key Ideas:

The sine law and cosine law can be used
to determine unknown side lengths and
angle measures in obtuse triangles.

The sine law and cosine law are used
with obtuse triangles in the same way
that they are used with acute triangles.

Be careful when using the sine law to
determine the measure of an angle. The
inverse sine of a ratio always gives an
acute angle, but the supplementary
angle has the same ratio. You must
decide whether the acute angle, 𝜃, or the
obtuse angle 180o – 𝜃 is the correct
angle for your triangle.
The measures of the angles determined using the cosine law are always correct.

4.2 Assignment: Nelson Foundations of Mathematics 11, Sec 4.2, pg. 170

Questions: 1, 2abc, 3ab, 4ab, 6, 9, 12
Section 4.3: The Ambiguous Case of the Sine Law
SSA is a case where you are given the values for two sides of a triangle and the angle opposite one of
the given sides:
The number of possible solutions to problems like this depends on the length of the sides given and
the height of the triangle.
 If the opposite side to the ___________________ angle is ___________________ than
the height, then there is _______ possible solution.
 If the opposite side is ___________________ than the height but still
___________________ than the other given side, then there are _____ possible triangles.
 If the opposite side is __________________ than the height and __________________
than the other given side, then there is ______ possible triangle.
 If the opposite side is the __________ length as the height there is _____ possible solution.
Example 1 – Given each SSA situation for ∆𝐴𝐵𝐶, determine how many triangles are possible.
(a)
(c)
∠𝐴 = 30o, a = 4 m, b = 12 m
∠𝐴 = 30o, a = 8 m, b = 12 m
(b)
(d)
∠𝐴 = 30o, a = 6 m, b = 12 m
∠𝐴 = 30o, a = 15 m, b = 12 m
Example 2 – in obtuse ∆ABC, B = 24o, b = 18cm and a = 22cm. calculate the measure of A to the
nearest degree. Is there more than one possible answer?
Example 3 – Martina and Carl are part of a team that is studying weather patterns. The team is about
to launch a weather balloon to collect data. Martina’s rope is 7.8 m long and makes an angle of 36.0o
with the ground. Carl’s rope is 5.9 m long. Assuming that Martina and Carl form a triangle in a vertical
plane with the weather balloon, what is the distance between Martina and Carl, to the nearest tenth of
a metre?
Step 1 – Draw the triangle. Is it an SSA situation? If so, find the height of the triangle.
Step 2 – Determine the number of possible triangles. Draw a sketch of all possible triangles.
Step 3 – Solve the problem for ALL situations in Step 2.
B
7.8
M
36.0o
B
h
Ѳ
x
7.8
5.9
C
M
36.0o
5.9
Ѳ
x
h
C
Situation 1:
Situation 2:
Remember: sin 𝜃 = sin (180o – Ѳ)
Star by finding C for each situation then solving for x
Now solve for B to determine x
Key Idea
 The ______________ case of the sine law may occur when you are given two side lengths and the
measure of an angle that is ____________ one of these sides (SSA). Depending on the measure of
the given angle and the lengths of the given sides, you may need to construct and solve _________,
______ or ______ triangles.
 In ∆𝐴𝐵𝐶, when h is the height of the triangle, and ∠𝐴 and the lengths of sides a
and b are given, and ∠𝑨 is _________, there are four possibilities to consider:
 In ∆𝐴𝐵𝐶, when h is the height of the triangle, and ∠𝐴 and the
lengths of sides a and b are given, and ∠𝑨 is ____________,
there are two possibilities to consider:
4.3 Assignment:pg. 183-185, Questions: 1ac, 2ace, 3, 4ad, 5, 6, 8