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PANDA – Cycle 2 Basic Trigonometry
Teaching Basic Trigonometry in Shanghai
Background Information
School Systems in England and Shanghai
Junior secondary
Key Stage 3
Key Stage 4
Year
Year 6
Year
Year
Year
Year
Year
Year
Year
11
(age
Year 7
2
3
4
5
8
9
10
(age
11)
16)
Primary
Junior secondary
Grade
Grade
Grade
1
Grade Grade Grade
5
Grade Grade Grade
9
(age
2
3
4
(age
6
7
8
(age
6)
11)
15)
Key Stage 1
England
Year 1
(age
5)
Shanghai
Primary
Key Stage 2
The first term of Grade 9 contains three topics: similar triangles, basic trigonometry and
quadratic function.
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PANDA – Cycle 2 Basic Trigonometry
Lesson sequence
1. The meaning of trigonometry (acute angles)1
Tan and Cot:
Start: how the ancient Egyptians measured the height of Egyptian Pyramid
Question 1: for a right-angled triangle, given an acute angle, the ratio of two right-angle sides
is a certain value?
Proof by similar triangles
Question 2: when the size of acute angle changes, would the ratio of the opposite side and the
adjacent side change correspondingly?
1
1 lesson for tan and cot; and 1 lesson for sin and cos
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PANDA – Cycle 2 Basic Trigonometry
Conclusion: Given a certain size of an acute angle, the ratio of opposite side and the adjacent
𝑡ℎ𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 𝑜𝑓 𝐴𝑛𝑔𝑙𝑒 𝐴
𝑎
side is certain. Definition of tangent: 𝑡𝑎𝑛𝐴 = 𝑡ℎ𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 𝑜𝑓 𝐴𝑛𝑔𝑙𝑒 𝐴 = 𝑏
Example 1: At Right-angled triangle ABC, ∠𝐶 = 90°, 𝐴𝐶 = 3, 𝐵𝐶 = 2. Find the value of
𝑡𝑎𝑛𝐴 and 𝑡𝑎𝑛𝐵.
Define the concept of cotangent
𝑐𝑜𝑡𝐴 =
𝑡ℎ𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 𝑜𝑓 𝐴𝑛𝑔𝑙𝑒 𝐴 𝑏
=
𝑡ℎ𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 𝑜𝑓 𝐴𝑛𝑔𝑙𝑒 𝐴 𝑎
𝑡𝑎𝑛𝐴 =
1
𝑐𝑜𝑡𝐴
Have a think: Right-angled triangle ABC, ∠𝐶 = 90°, how to express cotB? What is the
relationship of 𝑐𝑜𝑡𝐵 and 𝑡𝑎𝑛𝐴?
Example 2: At Right-angled triangle ABC, ∠𝐶 = 90°, 𝐵𝐶 = 4, 𝐴𝐵 = 5. Find the value of
𝑐𝑜𝑡𝐴 and 𝑐𝑜𝑡𝐵. (Pythagoras involved).
Exercise:
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PANDA – Cycle 2 Basic Trigonometry
Sin and Cos:
Back to last lesson graph which shows the ratio
Using the conclusions from similar triangles
to get the definition of
Sin and Cos.
Have a think: Right-angled triangle ABC, ∠𝐶 = 90°, how to express cosB? What is the
relationship of 𝑐𝑜𝑠𝐵 and 𝑠𝑖𝑛𝐴?
Definition: Trigonometric ratio - sin, cos, tan, cot
Example 1: At Right-angled triangle ABC, ∠𝐶 = 90°, 𝐴𝐵 = 17, 𝐵𝐶 = 8. Find the value of
𝑠𝑖𝑛𝐴 and 𝑐𝑜𝑠𝐴. (Pythagoras involved for cosA).
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PANDA – Cycle 2 Basic Trigonometry
Example 2: Given 𝑃(3, 4), find the value of tan, sin and cot of the angle 𝛼 between OP and
the positive x-axis. (Pythagoras involved for sin and cos).
3
Example 3: At Right-angled triangle ABC, ∠𝐶 = 90°, 𝐵𝐶 = 6, 𝑠𝑖𝑛𝐴 = 4. Find (1) the
length of AB; (2) the value of sinB.
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PANDA – Cycle 2 Basic Trigonometry
2. Find the values of Sin, Cos, Tan and Cot 2
Exact value
Reasoning the exact values for 45° first and then using midpoint to find the exact values for
30° 𝑎𝑛𝑑 60°
Note 1: English textbooks use equilateral triangle to proof
Note 2: Reasoning the exact values for 15° 𝑎𝑛𝑑 75° as extension
Example 1:
(Revision surds)
Solution:
2
1 lesson for the exact values of sin, cos, tan and cot for 30°, 45° and 60°;
and 1 lesson for angles by using calculator
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PANDA – Cycle 2 Basic Trigonometry
Exercise 4: As we know
Try to fill in
Find angles by calculator
Example 4:
Example 7: A Right-angled triangle ABC, ∠𝐶 = 90°, 𝐴𝐶 = 15, 𝐵𝐶 = 9, find the size of
angle A.
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PANDA – Cycle 2 Basic Trigonometry
3. Solve the right-angled triangle (Find the missing length and angle)3
Find the missing length and angle at right-angled triangle
A right-angled triangle contains three sides and two acute angles which has the following
relationship:
1) Three sides 𝑎2 + 𝑏 2 = 𝑐 2
2) Two acute angles ∠𝐴 + ∠𝐵 = 90°
3) Side and angels
Question: To find all values of sides and angles, what should be given?
Answer: either two sides or one side and one angle
Example 1 Right-angled triangle ABC, ∠𝐶 = 90°, ∠𝐵 = 38°, 𝑎 = 8, find the other sides and
angles.
Example 2 Right-angled triangle ABC, ∠𝐶 = 90°, 𝑐 = 7, 𝑎 = 5.28, find the other sides and
angles.
Find the missing length and angle at other types of triangles
Example 1 The isosceles triangle ABC, 𝐴𝐵 = 𝐴𝐶, ∠𝐴 = 45°, 𝐵𝐶 = 6, find the length of
AB and base angle.
3
1 lessons for the missing length and angle at right-angled triangle, and 1 lesson for
the missing length and angle at other types of triangles
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PANDA – Cycle 2 Basic Trigonometry
Have a try: And isosceles triangle ABC, 𝐴𝐵 = 𝐴𝐶 = 5, 𝐵𝐶 = 6, find the size of base angle.
Example 2 Triangle ABC, 𝐴𝐶 = 9, 𝐴𝐵 = 8.5, ∠𝐴 = 38°, find the height based on AC, and
the area of triangle ABC.
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PANDA – Cycle 2 Basic Trigonometry
4. Application of trigonometry in different real life context
Example 1 Point A is 10 metres away from the flagpole. By using the goniometer at Point A,
the angle of elevation is measured as 52°. The height of goniometer AD is 1.5 metres. Find
the height of the flagpole (1 decimal place).
Example 2 The distance between two buildings, CD, is 40 metres. Now in order to find the
height of building 2, BC (BC is perpendicular to CD), point A is chosen as observation place.
AD is parallel to BC, the angle of elevation from point A to point B is measured as 32°, the
depression angle from point A to point C is measured as 25°. Find the height of building 2
(whole number).
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PANDA – Cycle 2 Basic Trigonometry
Example 3 A ship is leaving the port A towards the east as the speed of 24 km per hour.
There is an island B, located as south by east 52° from the port A. After 20 mins, the ship
finds that the island B is located as its south. Find the distance between the island B and the
port A (1 decimal place).
Example 4 In order to measure the width of a river, two points B, C are chosen at one side of
the river, and point A is at the another side of the river. At triangle ABC, ∠𝐶 = 62°, ∠𝐵 =
49°, 𝐵𝐶 = 23.5 𝑚𝑒𝑡𝑟𝑒𝑠. Find the width of the river (1 decimal place).
Slope ratio 𝑖 =
ℎ
𝑙
= 𝑡𝑎𝑛𝛼, written as 1: 𝑚
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PANDA – Cycle 2 Basic Trigonometry
Example 8 The length from point O to the centre of the sphere is 50 cm. When the ball
reaches the highest position, E and F, the angle between OE and OF is 40°. Find the
differences between the highest and the lowest position.
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