Right Angled Triangles
邊玩邊學
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給老師： 為 Right Angled Triangles（Maths [CIE]、Extended）準備好可直接使用的課程投影片, 複習筆記, 圖表——用於你的課程，或把這個主題當成互動班級活動，讓學生以即時遊戲的方式進行。
課程筆記
Pythagoras Theorem
- Pythagoras' theorem: For a right-angled triangle with hypotenuse×c×and shorter sides×a×,×b×: a² + b² = c².
- The hypotenuse is the longest side, opposite the right angle.
- To find the hypotenuse: c = √(a² + b²) (add squares).
- To find a shorter side: a = √(c² – b²) (subtract squares).
- If the hypotenuse is shorter than another side, you have made a mistake.
- Pythagoras can be used in any shape that can be split into right-angled triangles (e.g., diagonal of a rectangle).
- In multi-step problems, leave intermediate answers as exact (e.g., √63) until the final step.
Right-angled triangle
SOHCAHTOA
- SOHCAHTOA is a mnemonic for trig ratios in right-angled triangles.
- sin θ = Opposite / Hypotenuse (SOH).
- cos θ = Adjacent / Hypotenuse (CAH).
- tan θ = Opposite / Adjacent (TOA).
- Label sides relative to angle θ: H (hypotenuse), O (opposite), A (adjacent).
- To find a missing side: choose the correct ratio, substitute known values, and rearrange.
- To find a missing angle: use inverse trig functions (e.g., θ = sin⁻¹(O/H)).
- Ensure calculator is in degree mode (D or Deg).
SOHCAHTOA
Angles of Elevation & Depression
- Angle of elevation: angle above horizontal when looking up at an object.
- Angle of depression: angle below horizontal when looking down at an object.
- These angles are measured from the horizontal to the line of sight.
- Right-angled trigonometry (often tan) is used to find distances or heights.
- Draw a clear diagram; use alternate angles if needed (e.g., angle of depression equals angle at base).
Angles of elevation and depression
Exact Trig Values
- Know exact values for 0°, 30°, 45°, 60°, 90°.
- sin 0° = 0, sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2, sin 90° = 1.
- cos 0° = 1, cos 30° = √3/2, cos 45° = √2/2, cos 60° = 1/2, cos 90° = 0.
- tan 0° = 0, tan 30° = √3/3, tan 45° = 1, tan 60° = √3, tan 90° undefined.
- Use special triangles (45°-45°-90° and 30°-60°-90°) to derive these values.
- In non-calculator questions, substitute exact values and simplify (e.g., cos45° = √2/2).
- Sketch the triangles or table at the start of the exam for quick reference.
Triangle for 30° and 60°
Problem Solving with Right-Angled Triangles
- Identify right-angled triangles within composite shapes (e.g., quadrilaterals, prisms, circles).
- The shortest distance from a point to a line is the perpendicular distance; form a right-angled triangle.
- In multi-step problems, find intermediate lengths using Pythagoras or trig before final answer.
- Round final answers as instructed: sides to 3 significant figures, angles to 1 decimal place.
- Check reasonableness: hypotenuse is longest side; angles in a triangle sum to 180°.
投影片
練習題
免費預覽——69 題中的 8 題。註冊即可查看全部。
1.In a right-angled triangle, the side opposite the right angle is called the
Easy- Ahypotenuse
- Badjacent
- Copposite
- Dbase
2.What is the value of sin 30°?
Easy- A0
- B1/2
- C√2/2
- D√3/2
3.Which trigonometric ratio is defined as opposite/hypotenuse?
Easy- Asine
- Bcosine
- Ctangent
- Dsecant
4.In a right-angled triangle with sides 3 cm and 4 cm, the length of the hypotenuse is
Easy- A5 cm
- B7 cm
- C1 cm
- D25 cm
5.A right-angled triangle has hypotenuse 10 cm and one leg 6 cm. The length of the other leg is
Medium- A8 cm
- B4 cm
- C√136 cm
- D16 cm
6.In triangle ABC, angle B = 90°, AB = 5 cm, BC = 12 cm. Find AC.
Medium- A13 cm
- B7 cm
- C17 cm
- D√119 cm
7.Calculate the angle x in a right-angled triangle where the opposite side is 5 cm and the adjacent side is 12 cm. Give your answer correct to 1 decimal place.
Medium- A22.6°
- B67.4°
- C24.6°
- D65.4°
8.A ladder 5 m long leans against a vertical wall. The foot of the ladder is 3 m from the wall. How high up the wall does the ladder reach?
Medium- A4 m
- B2 m
- C√34 m
- D8 m
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