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PANDA โ 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. 1|Page PANDA โ 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 2|Page PANDA โ Cycle 2 Basic Trigonometry Conclusion: Given a certain size of an acute angle, the ratio of opposite side and the adjacent ๐กโ๐ ๐๐๐๐๐ ๐๐ก๐ ๐ ๐๐๐ ๐๐ ๐ด๐๐๐๐ ๐ด ๐ side is certain. Definition of tangent: ๐ก๐๐๐ด = ๐กโ๐ ๐๐๐๐๐๐๐๐ก ๐ ๐๐๐ ๐๐ ๐ด๐๐๐๐ ๐ด = ๐ Example 1: At Right-angled triangle ABC, โ ๐ถ = 90°, ๐ด๐ถ = 3, ๐ต๐ถ = 2. Find the value of ๐ก๐๐๐ด and ๐ก๐๐๐ต. Define the concept of cotangent ๐๐๐ก๐ด = ๐กโ๐ ๐๐๐๐๐๐๐๐ก ๐ ๐๐๐ ๐๐ ๐ด๐๐๐๐ ๐ด ๐ = ๐กโ๐ ๐๐๐๐๐ ๐๐ก๐ ๐ ๐๐๐ ๐๐ ๐ด๐๐๐๐ ๐ด ๐ ๐ก๐๐๐ด = 1 ๐๐๐ก๐ด Have a think: Right-angled triangle ABC, โ ๐ถ = 90°, how to express cotB? What is the relationship of ๐๐๐ก๐ต and ๐ก๐๐๐ด? Example 2: At Right-angled triangle ABC, โ ๐ถ = 90°, ๐ต๐ถ = 4, ๐ด๐ต = 5. Find the value of ๐๐๐ก๐ด and ๐๐๐ก๐ต. (Pythagoras involved). Exercise: 3|Page PANDA โ 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, โ ๐ถ = 90°, how to express cosB? What is the relationship of ๐๐๐ ๐ต and ๐ ๐๐๐ด? Definition: Trigonometric ratio - sin, cos, tan, cot Example 1: At Right-angled triangle ABC, โ ๐ถ = 90°, ๐ด๐ต = 17, ๐ต๐ถ = 8. Find the value of ๐ ๐๐๐ด and ๐๐๐ ๐ด. (Pythagoras involved for cosA). 4|Page PANDA โ Cycle 2 Basic Trigonometry Example 2: Given ๐(3, 4), find the value of tan, sin and cot of the angle ๐ผ between OP and the positive x-axis. (Pythagoras involved for sin and cos). 3 Example 3: At Right-angled triangle ABC, โ ๐ถ = 90°, ๐ต๐ถ = 6, ๐ ๐๐๐ด = 4. Find (1) the length of AB; (2) the value of sinB. 5|Page PANDA โ 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° ๐๐๐ 60° Note 1: English textbooks use equilateral triangle to proof Note 2: Reasoning the exact values for 15° ๐๐๐ 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 6|Page PANDA โ 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, โ ๐ถ = 90°, ๐ด๐ถ = 15, ๐ต๐ถ = 9, find the size of angle A. 7|Page PANDA โ 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 ๐2 + ๐ 2 = ๐ 2 2) Two acute angles โ ๐ด + โ ๐ต = 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, โ ๐ถ = 90°, โ ๐ต = 38°, ๐ = 8, find the other sides and angles. Example 2 Right-angled triangle ABC, โ ๐ถ = 90°, ๐ = 7, ๐ = 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, ๐ด๐ต = ๐ด๐ถ, โ ๐ด = 45°, ๐ต๐ถ = 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 8|Page PANDA โ Cycle 2 Basic Trigonometry Have a try: And isosceles triangle ABC, ๐ด๐ต = ๐ด๐ถ = 5, ๐ต๐ถ = 6, find the size of base angle. Example 2 Triangle ABC, ๐ด๐ถ = 9, ๐ด๐ต = 8.5, โ ๐ด = 38°, find the height based on AC, and the area of triangle ABC. 9|Page PANDA โ 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). 10 | P a g e PANDA โ 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, โ ๐ถ = 62°, โ ๐ต = 49°, ๐ต๐ถ = 23.5 ๐๐๐ก๐๐๐ . Find the width of the river (1 decimal place). Slope ratio ๐ = โ ๐ = ๐ก๐๐๐ผ, written as 1: ๐ 11 | P a g e PANDA โ 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. 12 | P a g e