Chapter 2: Trigonometry - Mr. McDonald's Homepagemrmcdonaldshomepage.weebly.com/uploads/1/3/9/2/13921997/math_1… · Chapter 2: Trigonometry Section 2.1 32 The Tangent Ratio There

Chapter 2: Trigonometry Section 2.1

31

Chapter 2: Trigonometry

Section 2.1: The Tangent Ratio

Sides of a Right Triangle with Respect to a Reference Angle

Given a right triangle, we generally label its sides with respect to a chosen reference

angle. This is shown in the diagram below.

The side that touches the reference angle is referred to as the adjacent side (A).

The side that is directly across from the reference angle is referred to as the

opposite side (O).

The longest side of the triangle, the side across from the right angle, is the

hypotenuse (H).

Ex. Label the adj., opp., and hyp. according to angle 𝜃 in each triangle.

(a) (b)

𝜃

𝜃


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The Tangent Ratio

There is a relationship that exists between the chosen reference angle and the measure

of the sides that are opposite and adjacent to it. This is known as the Tangent Ratio

(Tan).

Determine the Tangent Ratios for Angles

Ex1. Determine Tan(D) and Tan(F) for the triangle

Ex2. Determine Tan(X) and Tan(Z) for the triangle

Tangent Ratio

If ∠A is an acute angle in a right triangle, then

tan(𝐴) =𝑙𝑒𝑛𝑔ℎ 𝑜𝑓 𝑠𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑡𝑜 ∠𝐴

𝑙𝑒𝑛𝑔ℎ 𝑜𝑓 𝑠𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 ∠𝐴

tan(𝐴) =𝑜𝑝𝑝

𝑎𝑑𝑗

tan(𝐴) =𝑂

𝐴 AB

C

Adjacent

Opposite

Hypotenuse

X

Y Z6

12

F

D E4

3


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Using the Tangent Ratio to Determine the Measure of an Angle

Ex1. Determine the measures of ∠G and ∠J to the nearest degree.

Ex2. Determine the measures of ∠K and ∠N to the nearest degree.

H

G I

45

M

K N

91 3


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Terminology:

Angle of Inclination:

The acute angle that is formed between a line segment and the horizon.

Using the Tangent Ratio to solve a Problem

Ex1. A 10 ft. ladder leans against the side of a building with its base 4 ft. from the wall.

What angle, to the nearest degree, does the ladder make with the ground?

Ex2. A support cable is anchored to the ground 5 m from the base of a telephone pole.

The cable is 19 m long. It is attached near the top of the pole. What angle, to the nearest

degree, does the cable make with the ground?

Practice Problems:

3,4,5,8,10,14,17 pg 75-77

Angle of Inclination


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Section 2.2: Using The Tangent Ratio to Determine Side Length

Determining the Length of a Side Given an Angle

Ex1. Determine the length of AB to the nearest centimetre.

Ex2. Determine the length of XY to the nearest foot.

B

CA

10 cm

27°

Z X

Y

5.0 ft

49°


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Ex3. Determine the length of EF to the nearest tenth of an inch.

Ex4. Determine the length of VX to the nearest tenth of a centimetre.

D

FE

3.5 in

24°

W

VX

7.2 cm

44°


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Word Problem Applications of Tangent

Ex1. A searchlight beam shines vertically on a cloud. At a horizontal distance of 250 m

from the line of sight to the cloud is 75°. Determine the height of the cloud to the nearest

metre.

Ex2. At a horizontal distance of 200 m from the base of an observation tower, the angle

between the ground and the line of sight to the top of the tower is 8°. How high is the

tower to the nearest metre?

Practice Problems

#3,4,5,6,7,8,9,10,13,14 pg 82-83


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Section 2.3: Using a Clinometer

Terminology:

Clinometer:

A device used to measure the angle between a horizontal line and the line of sight

to the top of an object.

NOTE 1: The reading on a clinometer gives the measurement of the angle formed

between the hypotenuse and the top of the object (vertical side).

NOTE 2: When calculating the height of an object using a clinometer, you must also add

on the height of the person making the measurements as a clinometer measures

according the user’s eye level.

Ex1. Determine the height of a tall tree if you are stood 10 m away from the base of the

tree and the clinometer you are using gives an angle measurement of 35°. Assume that

you are 1.7 m tall.

Ex2. Julia is 1.3 m tall. If she is stood 7.0 m away from a flag pole and uses a clinometer

to determine the angle between your line of sight and the top of the pole is 15°,

determine the height of the flag pole.

Practice Problem

#2 & 3 pg 86


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Section 2.4: The Sine and Cosine Ratios

Sometimes it is useful to have a mnemonic to help you to remember the three trig ratios

such as:

____________________________________________________________

____________________________________________________________

____________________________________________________________

Sine Ratio

If ∠𝜃 is an acute angle in a right triangle, then

Sin(𝜃) =𝑙𝑒𝑛𝑔ℎ 𝑜𝑓 𝑠𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑡𝑜 ∠𝜃

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

Sin(𝜃) =𝑜𝑝𝑝

ℎ𝑦𝑝

Sin(𝜃) =𝑂

𝐻

Cosine Ratio

If ∠𝜃 is an acute angle in a right triangle, then

Cos(𝜃) =𝑙𝑒𝑛𝑔ℎ 𝑜𝑓 𝑠𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 ∠𝜃

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

Cos(𝜃) =𝑎𝑑𝑗

ℎ𝑦𝑝

Cos(𝜃) =𝐴

𝐻

AB

C

Adjacent

Opposite

Hypotenuse

AB

C

Adjacent

Opposite

Hypotenuse

𝜃

𝜃


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Determining the Sine and Cosine Ratio for an Angle

Ex1. Determine 𝑆𝑖𝑛(𝐷) and 𝐶𝑜𝑠(𝐷) to the nearest hundredth

Ex2. Determine 𝑆𝑖𝑛(𝐺)𝑎𝑛𝑑 𝐶𝑜𝑠(𝐺) to the nearest hundredth

D

E

F13 cm

5cm12 cm

G

H J

17 cm

8cm

15 cm


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Using Sine or Cosine to Determine the Measure of an Angle

Ex1. Determine the measures of ∠G and ∠H to the nearest tenth of a degree.

Ex2. Determine the measures of ∠K and ∠M to the nearest tenth of a degree.

G

K

H14

6

M K

N

3

8


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Using Sine or Cosine to Determine an Unknown Side

Ex1. Determine the length of the unknown side in each triangle

(a)

(b)

(c)

(d)

56°

x

12 cm

41°

x

7 m

63°

x

18 ft

71°

x

9.5 in


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Terminology:

Angle of Elevation

The angle formed between the ground and the line of sight from an observer.

Angle of Depression

The angle formed between a horizon and the line of sight from an observer.

NOTE: The angle of elevation in a diagram will always equal the angle of depression.

Using Sine or Cosine to Solve a Problem

Ex1. A water bomber is flying at an altitude of 5000 ft. The plane’s radar shows that it is

a horizontal distance of 8000 ft. from the target site. What is the angle of elevation of

the plane measured from the target site, to the nearest degree?

Angle of Elevation

Angle of Depression


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Ex2. An observer is sitting on a dock watching a float plane in Vancouver harbour. At a

certain time, the plane is 300 m above the water and 430 m from the observer.

Determine the angle of elevation of the plane measured from the observer, to the

nearest degree.

Ex3. A surveyor made the measurements shown in the diagram. How could the surveyor

determine the distance from the transit to the survey pole to the nearest hundredth of a

metre?

Survey Pole

Survey Stake

Transit 20.86 m

67.3°


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Ex4. From a radar station, the angle of elevation of an approaching airplane is 32.5°.

The horizontal distance between the plane and the radar station is 35.6 km. How far is

the plane diagonally from the radar station to the nearest tenth of a kilometer?

Practice Questions

5, 7, 8, 10, 12 pg 95-96

3, 4, 5, 6, 7, 8 pg 101-102


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Section 2.5: Applying Trig Ratios in Multistep Questions

Solving a Right Triangle

Ex1. Solve △XYZ. Give the measures to the nearest tenth.

Ex2. Solve △MNK. Give measurements to the nearest tenth.

6.0 cm 10.0 cm

X

Y

Z

27°

7.0 cm

K

M

N


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Calculate a Side Length Using More than One Triangle

Ex1. Calculate the length of CD to the nearest tenth of a centimetre.

Ex2. Calculate the length of XY to the nearest tenth of a centimetre.

28°

36°

4.2 cmA D

B

C

22°

32°

8.4 cm

X W Z

Y


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Solving a Problem with Triangles in the Same Place

Ex1. From the top of a 20 m high building, a surveyor measured the angle of elevation of

the top of another building to be 30° and the angle of depression to the base of that

building to be 15°. The surveyor sketched this plan of her measurements. Determine the

height of the taller building to the nearest tenth of a metre.

Ex2. A surveyor stands at the window on the 9th floor of an office tower. He uses a

clinometer to measure the angles of elevation to be 31° to the top and depression to be

42° to the base of a taller building. Determine the height of the taller building to the

nearest tenth of a metre.

ShorterBuilding

TallerBuilding

20m

15°

30°

ShorterBuilding

TallerBuilding31°

42°

39 m


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Ex3. A tourist at Point Amour sees a fishing boat at an angle of depression of 23˚ and a

sailboat at an angle of depression of 9˚. If the tourist is 33.5 m above the water,

determine how far apart the two vessels are.

Three Dimensional Problem

Ex. From the top of a 90 ft observer tower, a fire ranger observes one fire due west of the

tower at an angle of depression of 5°, and another fire due south of the tower at an angle

of depression of 2°. How far apart are the fires to the nearest foot?

Practice Questions

#3,4,6,8,9,10,11,13,14 pg 118-20

2°5°

90 ft

Fire 1

Fire 2

Tower

Light House

Fishing Boat

Sail Boat

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Chapter 2: Trigonometry - Mr. McDonald's Homepagemrmcdonaldshomepage.weebly.com/uploads/1/3/9/2/13921997/math_1… · Chapter 2: Trigonometry Section 2.1 32 The Tangent Ratio There