Lesson Notes By Weeks and Term v5 - Grade 9

Lesson Video

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Performance objectives

• Apply the Theorem of Pythagoras to solve 2D problems.
• Identify hypotenuse, opposite, and adjacent sides.
• Calculate sine, cosine, and tangent ratios.
• Use trig ratios to find unknown side lengths.
• Use trig ratios to find unknown angles.

Lesson summary

This week, we delve deeper into the exciting world of Measurement and Trigonometry. Specifically, we will focus on applying our knowledge of the Theorem of Pythagoras to solve problems in two dimensions and introduce the trigonometric ratios of sine, cosine, and tangent for acute angles. This is incredibly important because measurement and trigonometry are fundamental tools used in various fields like construction, surveying, navigation, engineering, and even everyday problem-solving such as determining the height of a building or calculating the distance across a river.

Lesson notes

2.1 Revisiting the Theorem of Pythagoras The Theorem of Pythagoras states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (legs). Mathematically, this is expressed as: a² + b² = c² where: c is the length of the hypotenuse a and b are the lengths of the other two sides (legs)

Example 1: A farmer in KwaZulu-Natal wants to build a rectangular kraal for his goats. He decides to use the diagonal of the rectangle to ensure that the corners are perfectly right angles. The kraal will be 12 meters long and 5 meters wide. How long should the diagonal be?

Solution: The diagonal forms the hypotenuse of a right-angled triangle with sides 12m and 5m.

Using the Theorem of Pythagoras: c² = a² + b² c² = 12² + 5² c² = 144 + 25 c² = 169 c = √169 c = 13 meters Therefore, the diagonal should be 13 meters long.

Example 2: A ladder 6 meters long leans against a wall. The foot of the ladder is 2 meters away from the base of the wall. How high up the wall does the ladder reach?

Solution: The ladder forms the hypotenuse, the wall is one side, and the distance from the wall is the other side. Let 'h' be the height the ladder reaches. 6² = h² + 2² 36 = h² + 4 h² = 36 - 4 h² = 32 h = √32 h ≈ 5.66 meters The ladder reaches approximately 5.66 meters up the wall. 2.2 Introducing Trigonometric Ratios Trigonometry deals with the relationships between the angles and sides of triangles. For right-angled triangles, we define three fundamental trigonometric ratios for an acute angle (an angle less than 90 degrees): Sine (sin): sin(θ) = Opposite / Hypotenuse Cosine (cos): cos(θ) = Adjacent / Hypotenuse Tangent (tan): tan(θ) = Opposite / Adjacent Where: θ (theta) represents the angle.

Opposite: The side opposite to the angle θ.

Adjacent: The side adjacent to (next to) the angle θ (and not the hypotenuse).

Hypotenuse: The side opposite the right angle (always the longest side). Important

Note: These ratios are only defined for angles within a right-angled triangle. SOH CAH TOA is a useful mnemonic to remember these ratios.

Example 3: Consider a right-angled triangle ABC, where angle B is the right angle. Let angle A be θ. If AB = 4cm, BC = 3cm and AC = 5cm (hypotenuse). Find sin(θ), cos(θ), and tan(θ).

Solution: Opposite side to angle A (θ) = BC = 3cm Adjacent side to angle A (θ) = AB = 4cm Hypotenuse = AC = 5cm Therefore: sin(θ) = Opposite / Hypotenuse = 3/5 = 0.6 cos(θ) = Adjacent / Hypotenuse = 4/5 = 0.8 tan(θ) = Opposite / Adjacent = 3/4 = 0.75 2.3 Using Trigonometric Ratios to Find Unknown Sides If we know one acute angle and the length of one side in a right-angled triangle, we can use trigonometric ratios to find the lengths of the other sides.

Example 4: A flagpole casts a shadow of 8 meters long. The angle of elevation from the tip of the shadow to the top of the flagpole is 60°. Calculate the height of the flagpole.

Solution: Let 'h' be the height of the flagpole. Angle of elevation = 60° Adjacent side = 8m (length of the shadow) Opposite side = h (height of the flagpole)

We can use the tangent ratio: tan(60°) = Opposite / Adjacent tan(60°) = h / 8 h = 8 * tan(60°) h ≈ 8 * 1.732 h ≈ 13.86 meters Therefore, the height of the flagpole is approximately 13.86 meters. 2.4 Using Trigonometric Ratios to Find Unknown Angles If we know the lengths of two sides in a right-angled triangle, we can use trigonometric ratios and the inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) to find the measure of the acute angles. Your calculator will have these as "inv sin", "inv cos", or "inv tan" functions (often accessed by pressing the SHIFT or 2nd function key).

Example 5: In a right-angled triangle, the opposite side is 5 cm and the hypotenuse is 13 cm. Calculate the size of the angle opposite the 5cm side.

Solution: Let the angle be θ. Opposite side = 5cm Hypotenuse = 13cm We can use the sine ratio: sin(θ) = Opposite / Hypotenuse sin(θ) = 5/13 θ = sin⁻¹(5/13) θ ≈ 22.62° Therefore, the angle is approximately 22.62 degrees. Guided Practice (With Solutions)

Question 1: A rectangular field is 30 meters long and the diagonal is 50 meters long. Calculate the width of the field.

Solution: The diagonal forms the hypotenuse. Let 'w' be the width. 50² = 30² + w² 2500 = 900 + w² w² = 1600 w = √1600 w = 40 meters The width of the field is 40 meters. We used the Pythagorean theorem to relate the sides and the diagonal and solved for the unknown width.

Question 2: A ramp is built to provide access to a building. The ramp is 5 meters long and rises to a height of 0.5 meters. Calculate the angle of inclination of the ramp.

Solution: The ramp is the hypotenuse, and the height is the opposite side. Let the angle of inclination be θ. sin(θ) = Opposite / Hypotenuse sin(θ) = 0.5 / 5 sin(θ) = 0.1 θ = sin⁻¹(0.1) θ ≈ 5.74° The angle of inclination of the ramp is approximately 5.74 degrees.