2020 MS Trigonometry Notes Benjamin Odgers PDF

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Chapter 4

Trigonometry 4A Trigonometric Ratios (pg. 42) 4B Angles of Elevation and Depression (pg. 43) 4C Bearings (pg. 44) 4D Trigonometry and Obtuse Angles (pg. 45) 4E Area of a Triangle (pg. 46) 4F The Sine Rule (pg. 47) 4G The Cosine Rule (pg. 48) 4H Trigonometric Problems (pg. 49) 4I Radial Survey (pg. 50)

Written by Benjamin Odgers Maths Teacher B Teaching / B Science The following theory booklet lines up with the Cambridge Year 12 NSW Standard Mathematics 2 Textbook. This can be found using the following link: https://www.cambridge.edu.au/education/titles/CambridgeMATHS-Stage-6-Mathematics-Standard-2-Year-12-printand-interactive-textbook-powered-by-HOTmaths/#.XYgHTUszaUk

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4A Trigonometric Ratios https://youtu.be/FPyfhNXjvnU • Label the sides (hypotenuse, opposite, adjacent) • SOHCAHTOA • Slide, Switch or Invert

Hypotenuse Opposite 𝜃 Adjacent

A

H

T

O

A

hypotenuse

tangent

opposite

adjacent

opposite

C

adjacent

sine

H

cosine

O

hypotenuse

S

Example 1 – Slide or Switch https://youtu.be/TiYKoCkqfTU Calculate the value of the pronumeral in each right-angled triangle below, correct to 2 decimal places. a (a) (b) 53.8°

a 29°

13.8cm 6.1mm

(c)

(d) 23m

23.2cm

62°11′25′′

32°45′ b

b

Example 2 – Invert https://youtu.be/WVbJ_vjEZ1E (a) Calculate the value of θ, correct to 1 decimal place.

(b)

Calculate the value of θ, correct to the nearest minute. 2.2m

θ θ 2.4cm

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6.7cm

3m

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4B Angles of Elevation and Depression • Focus on the triangle first • Then focus on solving the problem

Angle of Elevation (going up)

Angle of Depression (going down)

Example 1 – Angle of Elevation https://youtu.be/_o4WkBSl9EU Jane needed to measure the height of one of the trees on her property. While standing 20 metres from the base of the tree she measured the angle of elevation from the ground to the top of the tree at 56°. What is the height of the tree?

Example 2 – Angle of Depression https://youtu.be/VH66a72284Y Grant is standing on a cliff that is 78 metres above sea level. He is looking at a boat that is 800 metres from the base of the cliff. Calculate the angle of depression of the boat from the cliff correct to the nearest minute.

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000° N

4C Bearings Compass Bearings and True Bearings • Compass bearings are measured from North or South • True bearings are measured clockwise from due North • True bearings are written using 3 digits

W 270°

E 090°

Example 1 https://youtu.be/TRmG940tyxc By referring to the diagrams below, give the compass and true bearings of: a) A from O N

Compass bearing:

N

A 40°

True bearing: b) B from O

S 180°

O

W

E

O

W

E

25°

B

Compass bearing: S

S

True bearing: c) C from O N

D

N

Compass bearing: 38°

True bearing:

O

W 67°

d) D from O Compass bearing:

C

S

E

W

O

E

S

True bearing: Example 2 https://youtu.be/VzFsWyT5l34 A helicopter leaves its starting point (O) and travels on a bearing of 145° for 70 kilometres. How far east has the helicopter travelled from its starting point?

Example 3 https://youtu.be/c_keS8lubRA A drone leaves its starting point (O) and flies 230 metres north, then 150 metres west. What bearing must the drone travel in order to take the shortest route back to the starting point, correct to the nearest degree?

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4D Trigonometry and Obtuse Angles https://youtu.be/Hx0rTKlVi8Y If we take two supplementary angles (angles that add up to 180°) we notice some very important trigonometric properties. These properties are explored below. sin 30° = sin 150° 150°

Acute Angles (𝟎° to 𝟗𝟎°) sin 30° = ___________ cos 30° = ___________ tan 30° = ___________

sin 𝜃 = sin(180 − 𝜃)

30°

Obtuse Angles (𝟗𝟎° to 18𝟎°) sin 150° = ___________ cos 150° = ___________ tan 150° = ___________

What happens when we find the trigonometric ratios of supplementary angles?

Example 1 https://youtu.be/tN8KxX1lBKc Calculate the value of the obtuse angle θ correct to the nearest degree. (a) sin 𝜃 = 0.5 (b) cos 𝜃 = −0.5

(c)

sin 𝜃 = 0.632

(d)

tan 𝜃 = −1

Example 2 https://youtu.be/gaCGU9mTA58 Calculate the value of the obtuse angle θ correct to the nearest minute. (a) (b) cos 𝜃 = −0.128 √2 sin 𝜃 = 2

(c)

tan 𝜃 = −3.5

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(d)

tan 𝜃 = −

√3 3

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4E Area of a Triangle https://youtu.be/_y3UEWS8by0 The following formula is used to calculate the area of a triangle using the length of two sides and the included angle. C

1 𝐴 = 𝑎𝑏 sin 𝐶 2

a

b

A

c

B

Example 1 https://youtu.be/4yr4zY43C6c Calculate the area of the following triangle, correct to 1 decimal place. 35.4 m 73° 28.3 m

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4F The Sine Rule https://youtu.be/6nHcYNJtuPk The sine rule can be used to calculate the size of angles and sides on non-right triangles. 𝑎 sin 𝐴

=

𝑏 𝑐 = sin 𝐶 sin 𝐵

C a

b

sin 𝐴 sin 𝐵 sin 𝐶 = = 𝑐 𝑎 𝑏

A

B

c

Example 1 https://youtu.be/tHVqAgwbGA4 Use the sine rule to calculate the value of x in the following triangle, correct to 1 decimal place.

9 cm

89° 41° x

Example 2 https://youtu.be/v6nclljtbv4 Use the sine rule to calculate the value of 𝜃 in the following triangles, correct to the nearest degree. (a) (b) θ

13 cm

14 cm

40.1 mm

48°

θ

60°

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53.1 mm

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4G The Cosine Rule https://youtu.be/llDVUn5BuAs The cosine rule can be used to calculate the size of angles and sides on non-right triangles. 𝑐 2 = 𝑎2 + 𝑏2 − 2𝑎𝑏 cos 𝐶

C

or cos 𝐶 =

𝑎 2 + 𝑏 2 − 𝑐2 2𝑎𝑏

a

b

A

B

c

Example 1 https://youtu.be/28SvbeYkkEc Use the cosine rule to calculate the value of x in the following triangle, correct to 2 decimal places. 8 cm 61° 12 cm

x

Example 2 https://youtu.be/0_uSNTSzljQ Use the cosine rule to calculate the value of 𝜃 in the following triangle, correct to the nearest minute. 24 m 𝜃 23 m

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20 m

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4H Trigonometric Problems https://youtu.be/uVHKkYbaH8I Sine Rule

Cosine Rule

𝑎 𝑏 𝑐 = = sin 𝐴 sin 𝐵 sin 𝐶

𝑐 2 = 𝑎2 + 𝑏2 − 2𝑎𝑏 cos 𝐶

C a

b

or or sin 𝐴 sin 𝐵 sin 𝐶 = = 𝑐 𝑎 𝑏

𝑎 2 + 𝑏 2 − 𝑐2 cos 𝐶 = 2𝑎𝑏

A

c

B

Example 1 https://youtu.be/SO8CPQR5OVM A helicopter leaves the launch pad (O) and travels due north for 52 kilometres. It then travels for 71 kilometres at a bearing of 110°. The helicopter then takes the shortest route back to the starting point. How far must the helicopter travel to get back to its starting point, correct to the nearest kilometre?

Example 2 https://youtu.be/rXpUezfycGo Calculate the length of x in the diagram below.

𝑥 65°

43° 5m

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4I Radial Survey A radial survey is used to calculate the area of a piece of land. Usually a surveyor will stand at a central point and measure the distance from that point to each corner. The surveyor also measures the angle between each line. Example 1 https://youtu.be/K-a6OxUxU-g Calculate the area of the following block of land. Give your solution correct to 1 decimal place.

100

135° 125°

Example 2 https://youtu.be/tHyqo6omxvU A block of land is in the shape of a quadrilateral. Its perimeter is made of 4 straight edges that connect at 4 vertices labelled A, B, C and O below. A surveyor decides to stand at one vertex (O) and measure the distance and bearing to the other 3 vertices (as shown in the diagram below left). Calculate the perimeter of the block of land. N

W

A (043°)

A

E

O 112°)

C (205°)

S

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C

B

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