📐 Part 1: Understanding Trigonometric Ratios
1. In a right-angled triangle, what does SOH CAH TOA help us remember?
The three types of angles in triangles
The formulas for sine, cosine, and tangent ratios
The Pythagorean theorem
How to measure triangle sides
2. For angle θ in a right-angled triangle, sin θ equals:
opposite ÷ hypotenuse
adjacent ÷ hypotenuse
opposite ÷ adjacent
hypotenuse ÷ opposite
3. In a right-angled triangle with sides 3 cm, 4 cm, and 5 cm, what is cos of the angle opposite the 3 cm side?
4. Label the sides of this right-angled triangle relative to angle A:
Write: opposite, adjacent, and hypotenuse on the appropriate sides.
🧮 Part 2: Calculating Trigonometric Ratios
5. A right-angled triangle has a hypotenuse of 10 cm and one side of 6 cm. Calculate the missing side length, then find sin, cos, and tan for the angle opposite the 6 cm side.
Missing side: _______ cm
sin θ = _______
cos θ = _______
tan θ = _______
6. Using your calculator, find the following values (round to 3 decimal places):
sin 30° = _______
cos 45° = _______
tan 60° = _______
7. A surveyor measures a building's height by standing 20 metres away and measuring an angle of elevation of 35°. Which trigonometric ratio would you use to find the building's height?
Explain your choice:
🌊 Part 3: Real-World Applications
8. Traditional Māori navigators used the angle of stars above the horizon to determine their position at sea. If a navigator observes a star at an angle of 25° above the horizon, and knows they are 50 kilometres from their destination island, approximately how high is the star above sea level at that distance?
Height above sea level: _______ kilometres
9. Explain in your own words how trigonometric ratios might be useful in:
a) Navigation:
b) Construction or surveying:
10. Reflection: What was the most challenging part about learning trigonometric ratios today, and what strategy helped you understand them better?
✅ Answer Key
1. SOH CAH TOA helps us remember the formulas for sine, cosine, and tangent ratios.
2. sin θ equals opposite ÷ hypotenuse.
3. cos of the angle opposite the 3 cm side is 4/5.
4. Label the sides as follows: opposite (3 cm), adjacent (4 cm), hypotenuse (5 cm).
5. Missing side is 8 cm (using Pythagorean theorem: 10² - 6² = 8²).
sin θ = 6/10 = 0.6, cos θ = 8/10 = 0.8, tan θ = 6/8 = 0.75.
6. sin 30° = 0.500, cos 45° = 0.707, tan 60° = 1.732.
7. Use tangent (tan = opposite/adjacent). Explanation: You can find the height using the tangent ratio since you have the angle and the distance from the building.
8. Height above sea level can be calculated using tan(25°) = height/50 km, leading to height ≈ 23.5 km.
9. a) Navigation: Trigonometric ratios help determine distances and angles for accurate navigation.
b) Construction or surveying: They are used to calculate heights and distances when direct measurement is not possible.
10. Reflection: Answers will vary based on individual experiences.
