Download Sine and Cosine Laws

Name:__________________________
Trigonometry Unit:
#
1.
Assignment
Sine Law pg 2
2.
Assignment #1 pg 6
3.
Cosine Law pg 9
4.
Assignment #2 pg 11
5.
Assignment #3 pg 15
Completed?
Comments
Trig Test:_________________________
Page 1 of 19
Lesson #1
Sine Law
How do we label triangles??
Trigonometry can be used in everyday life to solve for angles and missing sides for
many problems. You may remember SOH CAH TOA from grade 10, this was an
acronym used to solve for sides and angles of a right angled triangle.
Example #1.
You can also use the Pythagorean Theorem to solve for a missing side.
a2 + b2 = c2
Example #2. Solve for c.
Page 2 of 19
Example #3. Solve for a.
The Sine Law can be used with oblique triangles (those that do not contain a 900
angle) to solve for unknown sides and angles.
The Sine Law states that the ratio of the Sine of an angle compared to the
measure of the side opposite that angle is the same for the entire triangle. The
inverse of this ratio is also true.
sin A sin B sin C


a
b
c
a
b
c


sin A sin B sin C
or
Where, side a is opposite angle A, side b is opposite angle B, etc.
A
c
B
b
a
C
Page 3 of 19
Example 1: Find the measures of side a and b and angle C.
1100
59.6 m
A
B
23.10
C
Example 2: Find the measures of angles D and F and side f.
D
13 cm
F
E
45°
14 cm
Page 4 of 19
Example 3: Boats are anchored at positions J, K and M on a lake. Boats J and K are
80 m apart and J and M are 110 m apart. The angle between the lines of sight from
K to J, and K to M is 1200. What is the angle between the lines of sight from J to K
and J to M? How far is it from K to M?
Page 5 of 19
Assignment #1. Sine Law
1. Given ΔABC, label its sides and angles with its proper letters.
2. Given that ∆ABC is a right angled triangle, side a = 13, side b = 12, and side c
is the hypotenuse, use the Pythagorean Theorem to solve for side c.
3. Given that ΔRST is a right angled triangle, r = 7, s = 11. Use the
Pythagorean Theorem to solve for t.
r=7
s = 11
t=?
4. Given Δ JKL, if ∠J = 34°, ∠K = 72°, what is ∠L?
Page 6 of 19
5. Find side b, given ∠B = 40°, ∠A = 70°, and a = 6 cm.
A
B
C
6. Determine the measure of ∠X.
X
11.7cm
5.6cm
108°
Z
Y
NOTE: For the following questions, make sure to draw a diagram to help you
with the question.
7. From two points A and B on the same side of a pond, the distances to a point
C, on the opposite side of the pond were measured and found to be 2000 m
and 1474 m respectively. If ∠ A = 25° and ∠ B = 35°, find the distance AB
across the bay.
Page 7 of 19
8. A children’s slide is 10 feet long and inclines to 43° from the ground. The
ladder is 8 feet long. What angle does the ladder make with the slide?
NOTE: Make sure to draw a diagram.
9. Mr. Krahn is trying to make a winter fishing shack with the following
dimensions.
a. How long is the shorter section of the roof?
4.5 m
35°
65°
b. How wide is the shed, to the nearest tenth of a meter?
Page 8 of 19
Lesson #2: Cosine Law
The Sine Law allows you to solve for triangles where 2 sides and a corresponding
angle are known, or 2 angles and a corresponding side.
The cosine law allows you to solve for:

the third side of the triangle if you know 2 sides of a triangle and the angle
that is formed between these two sides. (SAS or side – angle – side)

any angle if you know the three side lengths of the triangle.
a2 = b2 + c2 – 2bc cos A
 cos A = b2 + c2 – a2
(2bc)
 When solving triangles:

Check for right angles (900).  Use basic trigonometric ratio’s (SOH CAH TOA)

Check for Sine Law Ratios (a side and an opposite angle).  Use Sine Law.

If none of the above possibilities exist:  Use Cosine Law.
Page 9 of 19
Example: Find the measure of side a and angles B and C.
1. Right angles?
2. Sine-Law ratio’s? (opposites?)
3. Use Cosine.
A
55°
14 cm
18 cm
C
B
Page 10 of 19
Assignment #2: Cosine Law
1. Given ∆ABC. Solve for side a.
A
68°
c = 350
b= 475
B
C
2. Given ∆ABC. Solve for ∠A.
A
c = 55
b = 75
B
a = 70
C
3. From a lighthouse, a cruise ship can be seen 8.3 km away and a freighter can also be
seen 12.5 km away. How far away is the cruise ship from the freighter if the angle
between the lines of observation are 68°?
Lighthouse
8.3
Cruise Ship
68°
12.5
?
Freighter
Page 11 of 19
4. Solve for all the interior angles.
A
c = 18 cm
B
b = 20 cm
a = 19 cm
C
NOTE: For the following questions, make sure to draw a diagram to help you with
the question.
5. At a provincial park, there is a sign, a reception area, and a picnic area. The
reception area is 350 m away from the picnic area, the picnic area is 475 m away
from the sign. From the picnic area, the angle between the 2 lines of sight for the
reception area and the sign is 64°. How far apart is the sign from the reception
area?
Page 12 of 19
6.
An Art Gallery is in the shape of a triangle. Two of the walls are 114 m and 61 m in
length. The angle between these 2 walls is 72°.
a. How long is the 3rd wall?
b. What are the angles of the other 2 corners of the triangle?
Page 13 of 19
7. Construction has been started on a building as shown by the diagram.
12 ft
Pier
10 ft
Braces
a. What is the length of each brace?
b. What is the angle between each brace?
Page 14 of 19
Assignment #3: Applications of Sine & Cosine Laws
When to use Cosine vs Sine Law??
Complete the following chart.
Information Give
Measurement to be
Determined
Sine Law or Cosine Law
2 sides and the angle opposite one side
Angle
X
Y
2 angles and a side
Side
X
2 sides and the
contained angle
3 sides
Y
X
X
y
Side
Angle
z
Diagrams are not to scale. If no diagram is given, sketch one to represent the situation
before completing the exercise. Express all lengths to the nearest tenth and all angles to
the nearest degree.
1. For each triangle, determine the indicated measures.
Page 15 of 19
2.
From a certain point, the angle of elevation to the top of a church steeple is 9°. At
a point 100 m closer to the steeple, the angle of elevation is 15°. Calculate the
height of the steeple.
Page 16 of 19
3.
A tower is supported by two guy wires attached to the top of the tower and fixed
to the ground on opposite sides of the tower 27 m apart. One wire is 19.3 m long
and meets the ground at an angle of 53°.
a) What is the height of the tower?
b) What is the length of the second wire?
c) What angle does the second wire make with the ground?
4.
A triangular park has sides of length 200 m, 155 m and 172 m.
a)
Determine Angle A.
b)
Determine h.
Page 17 of 19
c)
5.
6.
Calculate the area of the park. (area of a triangle = ½base X height)
To determine the height of a cliff, a surveyor measured the angle of elevation of
the top of the cliff from a point away from the base to be 45°. He then moved 20
m further away from the base of the cliff and found the angle of elevation to the
top to be 37°. Determine the height of the cliff.
The end of a lean-to for cattle is in the shape of an obtuse triangle as shown below.
a)
Determine the length of the roof.
b)
Determine the angle that the roof of the shed makes with the ground.
Page 18 of 19
7.
In the design of a ski chalet, the slant of the roof must be steep enough for the
snow to slide off. An architect originally designed the roof to span 45 feet with
slanted sides of 36 ft and 30 ft. He decided it would be better to modify the roof
by increasing the measure of the smaller angle by 10° thus increasing the length of
the side opposite that angle.
a)
What is the new angle measure?
b)
What is the new length of this side?
Page 19 of 19