Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Define sine, cosine, and tangent ratios in a right-angled triangle.
• Calculate the values of these ratios.
• Use trig ratios to find unknown side lengths.
• Use trig ratios to find unknown angles.
• Solve problems involving angles of elevation and depression.

Lesson summary

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. In Grade 10, we start building the foundation for more advanced trigonometric concepts. This week, we focus on understanding and applying the definitions of trigonometric ratios (sine, cosine, and tangent) in right-angled triangles to solve for unknown sides and angles. This is incredibly important because these skills are used in surveying, navigation, engineering, architecture, and many other fields. Imagine building a house – you need to know the angles and lengths of the roof supports to ensure it's stable!

Lesson notes

2.1 Trigonometric Ratios in a Right-Angled Triangle Consider a right-angled triangle ABC, where angle B is 90°. Let's focus on angle

A. Hypotenuse: The side opposite the right angle (the longest side). In triangle ABC, the hypotenuse is A

C. Opposite Side: The side opposite to the angle we are considering. For angle A, the opposite side is B

C. Adjacent Side: The side adjacent to the angle we are considering (not the hypotenuse). For angle A, the adjacent side is A

B. The three primary trigonometric ratios are defined as follows: Sine (sin): The ratio of the length of the opposite side to the length of the hypotenuse. sin A = Opposite / Hypotenuse = BC / AC Cosine (cos): The ratio of the length of the adjacent side to the length of the hypotenuse. cos A = Adjacent / Hypotenuse = AB / AC Tangent (tan): The ratio of the length of the opposite side to the length of the adjacent side. tan A = Opposite / Adjacent = BC / AB Mnemonic: A helpful way to remember these ratios is using the acronym SOH CAH TOA: SOH: Sine = Opposite / Hypotenuse CAH: Cosine = Adjacent / Hypotenuse TOA: Tangent = Opposite / Adjacent Important Notes: These ratios are only defined for acute angles (angles less than 90°) in right-angled triangles. The trigonometric ratios are just numbers; they don't have units. 2.2 Finding Trigonometric Ratios To find the value of a trigonometric ratio for a given angle, you need to know the lengths of the sides of the right-angled triangle.

Example 1: Consider a right-angled triangle PQR where angle Q = 90°, PQ = 3 cm, QR = 4 cm, and PR = 5 cm. Find sin P, cos P, and tan

P. Solution: Hypotenuse = PR = 5 cm Opposite (to angle P) = QR = 4 cm Adjacent (to angle P) = PQ = 3 cm sin P = Opposite / Hypotenuse = 4 / 5 = 0.8 cos P = Adjacent / Hypotenuse = 3 / 5 = 0.6 tan P = Opposite / Adjacent = 4 / 3 = 1.33 (rounded to two decimal places)

Example 2: In a right-angled triangle XYZ, where angle Y = 90°, XZ = 13 cm and XY = 5 cm. Find sin Z, cos Z, and tan

Z. First, use Pythagoras' theorem to find the length of Y

Z. Solution: Using Pythagoras' Theorem: XZ² = XY² + YZ² 13² = 5² + YZ² 169 = 25 + YZ² YZ² = 144 YZ = √144 = 12 cm Now we can calculate the trigonometric ratios: Hypotenuse = XZ = 13 cm Opposite (to angle Z) = XY = 5 cm Adjacent (to angle Z) = YZ = 12 cm sin Z = Opposite / Hypotenuse = 5 / 13 = 0.38 (rounded to two decimal places) cos Z = Adjacent / Hypotenuse = 12 / 13 = 0.92 (rounded to two decimal places) tan Z = Opposite / Adjacent = 5 / 12 = 0.42 (rounded to two decimal places) 2.3 Finding Unknown Sides and Angles We can use trigonometric ratios to find unknown sides or angles in a right-angled triangle if we know one side and an angle (other than the right angle) or two sides.

Example 3: Finding an Unknown Side A ladder leans against a wall, making an angle of 60° with the ground. The foot of the ladder is 2 meters away from the wall. How high up the wall does the ladder reach?

Solution: Let the height the ladder reaches be 'h' meters. We have the adjacent side (2 m) and need to find the opposite side (h). We know tan (angle) = Opposite / Adjacent tan 60° = h / 2 h = 2 tan 60° Using a calculator, tan 60° ≈ 1.732 h ≈ 2 1.732 = 3.464 meters Therefore, the ladder reaches approximately 3.46 meters up the wall.

Example 4: Finding an Unknown Angle The length of a shadow cast by a tree is 8 meters. The tree is 6 meters tall. What is the angle of elevation of the sun?

Solution: Angle of elevation is the angle between the ground and the line of sight to the top of the tree. We have the opposite side (6 m) and the adjacent side (8 m). We know tan (angle) = Opposite / Adjacent tan (angle) = 6 / 8 = 0.75 To find the angle, we use the inverse tangent function (arctan or tan⁻¹). angle = tan⁻¹(0.75) Using a calculator, angle ≈ 36.87° Therefore, the angle of elevation of the sun is approximately 36.87°. 2.4 Angles of Elevation and Depression Angle of Elevation: The angle formed between the horizontal line and the line of sight upwards to an object.

Angle of Depression: The angle formed between the horizontal line and the line of sight downwards to an object.

Key Point: The angle of elevation from point A to point B is equal to the angle of depression from point B to point A (alternate angles are equal when the horizontal lines are parallel). Guided Practice (With Solutions)

Question 1: In a right-angled triangle ABC, where angle B = 90°, AB = 8 cm and BC = 15 cm. Find sin A, cos A, and tan

A. Solution: First, find the hypotenuse AC using Pythagoras' Theorem: AC² = AB² + BC² = 8² + 15² = 64 + 225 =

2

8

9. Therefore, AC = √289 = 17 cm. sin A = Opposite / Hypotenuse = BC / AC = 15 / 17 cos A = Adjacent / Hypotenuse = AB / AC = 8 / 17 tan A = Opposite / Adjacent = BC / AB = 15 / 8

Commentary: This question reinforces the basic definitions of the trigonometric ratios and requires the use of Pythagoras' theorem to find the missing side.