2. Geometry & Trigonometry
• Geometry is a branch of mathematics that
investigates the measurement and
relationships of lines, points, and shapes.
Trigonometry is a more specific study of
the measurement and relationship of sides
and angles in a triangle.
3. Geometry & Trigonometry
• In this unit you will learn about various
properties and relationships in geometry
and then apply them as you solve real
world problems using geometry and
trigonometry. This will involve the
calculation of angles and side lengths in
various shapes including triangles.
4. Properties of Triangles
• You may recall a few types of triangles. A Right
Triangle has one angle that is 90°. An
Isosceles Triangle has at least two congruent
(equal) sides. If there are exactly two congruent
sides they are called legs, and its non-congruent
side is the base. Angles opposite the equal
sides are equal. In an Equilateral Triangle, all
three sides are congruent and each of the three
angles equal 60°.
8. The Sine Ratio
The ratio of the length of the side opposite an angle to the length of the
hypotenuse is very important in the study of trigonometry. It is referred to as the
sine of an angle.
9. The Cosine Ratio
The cosine of an angle is the ratio of the length of the side adjacent to a given
angle to the length of the hypotenuse.
The word adjacent means close or adjoining. The side adjacent to a particular
angle is the side of the triangle remaining after you label the hypotenuse and the
side opposite the angle.
11. The Tangent Ratio
The tangent of an angle is the ratio of the length
of the side opposite a given angle to the length
of the side adjacent to the given angle.
13. Sine, Cosine, and Tangent
These three ratios can be shortened to:
The symbol θ (theta) is often used in mathematics to refer to an unknown angle.
It is a letter of the Greek alphabet. It serves as a reminder of the contribution
that ancient Greece made to the study of mathematics.
Some students find it helpful to remember the trigonometric relationships of
sine, cosine, and tangent by the following mnemonic, SOH CAH TOA
pronounced “sock-a-toe-a”.
16. Practice
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Notebook Assignment
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Q. 1 - 7
17. Solving Trigonometry Problems –
Strategies
To solve word problems involving trigonometric ratios, note the following 7-step
procedure:
1. Read the problem carefully.
2. Identify the given and unknown information.
3. Draw a diagram and label it appropriately.
4. Choose and substitute into the appropriate trigonometric ratio.
5. Solve for the unknown.
6. Check your answer to see if it is reasonable.
7. Make a concluding statement.
19. Practice
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20. Applications of Pythagorus: Carpentry
The Pythagorean Theorem, a2 + b2 = c2, is used in construction. Carpenters,
when raising walls for a house, need to know if the walls are "square" to each
other and form right angles in the corners, before they are pinned down. One
way they do this is by measuring 3 feet along one wall and making a pencil
mark, then measuring 4 feet along the second wall and making another mark.
The carpenter will place their tape measure at a diagonal between these two
pencil marks and make adjustments to the walls until the tape measure reads 5
feet. The carpenter is applying the Pythagorean triple: 32 + 42 = 52.
21. Applications of Trigonometry: Navigation
A helicopter is 10 km due west of its base when it receives a call to pick up a
stranded hiker. The hiker is 15 km due north of the helicopter's present position.
Once the helicopter picks up the hiker, what is the measure of the angle
between the route to the base and its current route?
Sample Solution:
The helicopter's route to base is 34 degrees away from its current route.
22. Applications of Trigonometry:
Architecture
A right rectangular pyramid is made of 4 isosceles triangles with a square base.
The Great Pyramid in Egypt has a square base with side length 755 ft and a
height of 481 ft. Determine the lateral surface area to the nearest foot.
Sample Solution:
Calculate the outer triangle's slant height by using the
Pythagorean Theorem.
Half of the base is 755 ft/2 = 377.5 ft.
Then, calculate the area of this triangle.
Lastly, multiply the area by 4 triangles.
Area = 230 822 ft2 x 4
Area = 923 289.5 ft2
23. Practice
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24. Practice
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27. Properties of Polygons
A polygon is a figure formed by three or more segments (sides).
Convex polygons are polygons in which each interior angle measures less
than 180°. In other words, the polygon does not "cave" in on any side. In
Concave polygons, one or more interior angles may measure more than 180°.
Convex: Concave:
In a regular polygon, all sides are equal in
length and all angles are the same
measurement.
28. Interior Angle Properties of Polygons
The sum of all interior angles in a convex polygon can be found by using the
formula:
Sum = 180°(n − 2)
Where n = the number of sides of the polygon.
Example 1:
Find the measurement of all the interior angles in a regular polygon with 5 sides.
Sample Solution:
Sum = 180°(n − 2) and n = 5
Sum = 180°(5 − 2)
Sum = 180°(3)
Sum = 540°
29. Interior Angle Properties of Polygons
The sum of all interior angles in a convex polygon can be found by using the
formula:
Sum = 180°(n − 2) Where n = the number of sides of the polygon.
Example 2:
Find the measurement of one of the angles in the regular 5-sided polygon.
Sample Solution:
You found the sum was 540° for all 5 angles, so you divide the sum by 5:
30. Exterior Angle Properties of Polygons
The sum of all measures of the exterior angles in a regular polygon equals
360°. Therefore, to find the measure of one exterior angle, divide by n.
31. Exterior Angle Properties of Polygons
Example
Find the measurement of the exterior angle of a regular 5-sided polygon.
Sample Solution:
The exterior angle measures
It is important to note that you just discovered that a 5-sided polygon has interior
angles which each measure 108°.
Together, the interior angle (108°) and the exterior angle (72°) add up to 180°.
32. Practice
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33. Practice
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34. Law of Sines
The law of sines can be used to calculate the unknown side lengths and angle in
a non-right triangle when two angles and a side are known. It can also be used
when two sides and one of the angles across from a given side are known. The
lowercase letters represent the sides and the uppercase letters represent the
angles opposite those sides. Consider the following diagram:
36. Law of Sines
When you solve an oblique triangle you cannot use the Pythagorean Theorem
or SOH CAH TOA because there is no right angle. One method to solve these
triangles is to use the law of sines.
The law of sines, or the sine law, is an equation that relates the sides of any
triangle to the sine of its angles.
According to the law:
37. Using The Law of Sines to find a Missing Side
When you solve for a missing side in an oblique triangle, you should set up the
ratio with the sides on top of the law of sines formula. Then, when you solve for
the unknown side, it will be on top making the ratio easier to solve.
Example:
Calculate the length of side b in the following triangle.
38. Using The Law of Sines to find a Missing Side
Sample Solution:
If you were to label this diagram, you could say:
Notice that you do not need to label side c or Angle C as they are not needed in
this diagram. Also notice, that the ratios have been written with the sides on the
top. This will make solving for an unknown side easier.
Therefore, you will only use the part of the law of sines that uses the a-values
and the b-values. To solve for b, you will substitute the correct values in for each
variable, and then solve algebraically for the missing variable.
Therefore, side b equals 5.08 cm.
39. Using The Law of Sines to find a Missing Angle
Example:
Calculate Angle C in the following triangle.
40. Using The Law of Sines to find a Missing Side
Sample Solution:
The Law of Sines formula is still used, but it is best if you flip the fractions upside
down when solving for angles. To solve for an angle, substitute the values in for
each variable, solve algebraically, and then take the inverse sine (sin-1) to find
the angle.
41. Applications of the Law of Sines
Triangles are thought of as the strongest shape in construction because they
have the smallest number of sides and angles of any polygon.
Carpenters may use the law of sines when calculating the angle of roof peaks,
when building trusses to support a roof, or when constructing ramps and sloped
walkways. Calculating the sides and angles of a triangle can then help the
carpenter determine the area needed for surfacing, roofing, or for pouring
concrete.
42. Applications of the Law of Sines
Example 1:
Dan is building a skateboard ramp. He uses a mitre saw to cut a triangular piece
as a side brace to support the ramp. (A mitre saw can be adjusted to cut at
specific angles within a quarter of a degree).
The base of the ramp measures 1.87 metres. For a smooth dismount when
performing a stunt at the peak of the ramp, the ramp must measure 120° at the
peak.
At what angle should Dan cut the wood if the side opposite that angle measures
0.74 m?
43. Applications of the Law of Sines
Sample Solution:
Dan should cut the wood at an angle of 20°.
44. Applications of the Law of Sines
Trigonometry is applied in navigation, surveying, satellite operations, and naval
and aviation industries.
Cartographers (map designers) for example and land surveyors need specific
calculations for their profession which may involve the law of sines.
These calculations help many industries such as tourism, travel, and aviation
with the distance of trails, heights of mountains, and navigation.
45. Applications of the Law of Sines
Example 1:
Two hikers want to view some waterfalls. The direct
walking distance from their location to the waterfalls
is 2.5 kilometres. However, because of thick bush
and difficult terrain, the hikers turn at an angle of 22°
from their original path, hike for 2 km and then turn
135° towards the falls to complete their hike.
How much farther did they have to go because of
their detour?
46. Applications of the Law of Sines
Sample Solution:
To find the total length of the detour, you must find the missing side first.
The total length of the detour is 2 km + 1.3 km = 3.3 km.
3.3 km − 2.5 km = 0.8 km.
Thus, the hikers travelled 0.8 km farther.
47. Applications of the Law of Sines
Architecture, interior decorating, digital imaging, and music production are all
real life applications that employ trigonometry.
Builders of bridges and buildings use angles and trigonometry to engineer a
stable architectural structure.
When redecorating a family room, you may consider the angles and distance
between lights and speakers of a home theatre system in order to receive the
best sound and brightness for watching a movie. Sound engineers will calculate
these angles for staged concerts to give viewers an optimum experience.
48. Practice
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49. Practice
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52. Law of Cosines
The law of sines helped you to solve oblique triangles when you knew the
value of one side, its opposite angle, and one other part of the triangle.
However, when you are given an oblique triangle, you are not always given
an opposing pair — a side that matches with an opposite angle.
Law of Cosines:
a2 = b2 + c2 − 2bccos(A)
There are only two cases when you will have to use the
law of cosines:
•When you are given all three sides and no angles.
•When you are given two sides and the angle between
them.
53. Using The Law Of Cosines
To Find A Missing Side
Example:
Calculate side a in the following triangle.
54. Using The Law Of Cosines
To Find A Missing Side
Sample Solution:
To solve for side a, substitute in the correct values for each variable, and
then solve algebraically for the missing variable.
55. Using The Law Of Cosines
To Find A Missing Side
Sample Solution:
If you want to solve a triangle for side b or side c, you can change the
formula to start with b2 = or c2 = and adjust the other variables as long as the
formula ends with the same variable that you started with!
a2 = b2 + c2 − 2bccos(A) OR
b2 = a2 + c2 − 2accos(B) OR
c2 = a2 + b2 − 2abcos(C)
Please note that when you are solving a
question involving the law of cosines you need
to be very careful to follow the correct order of
operations. Ensure that you find the entire
product of 2bc(cosA) before you subtract from
the sum of b2 + c2.
56. Using The Law Of Cosines
To Find A Missing Angle
Example:
Determine the measure of ∠C.
57. Using The Law Of Cosines
To Find A Missing Angle
Sample Solution:
You can substitute the known values into c2 = a2 + b2 − 2abcos(C) and then
use your algebraic skills to rearrange the formula and solve for the angle.
However, there are easier versions of the Law of Cosines to solve for an
angle.
58. Practice
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59. Practice
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