Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Define and calculate sine, cosine, and tangent ratios.
• Use trig ratios to find unknown side lengths in right-angled triangles.
• Use trig ratios to find unknown angles in right-angled triangles.
• Solve practical problems involving angles of elevation and depression.

Lesson summary

Trigonometry is a fundamental branch of mathematics that explores the relationships between angles and sides of triangles. In Grade 10, we build upon the foundations laid in earlier grades by formalizing trigonometric ratios and using them to solve problems involving right-angled triangles. Week 6 is crucial as we delve deeper into applying these ratios to find unknown sides and angles. This understanding is not just confined to the classroom; it's essential for various fields like surveying, navigation, engineering, and even game development. Imagine building a house, planning a hiking route, or designing a bridge – all require a solid understanding of trigonometry.

Lesson notes

2.1 Trigonometric Ratios In a right-angled triangle, the trigonometric ratios relate an acute angle to the ratio of two of its sides. Let's consider a right-angled triangle ABC, where angle C is the right angle (90°). Let angle A be our reference angle (an acute angle).

Hypotenuse: The side opposite the right angle (longest side) – in our case, side c.

Opposite: The side opposite to angle A – in our case, side a.

Adjacent: The side adjacent to angle A (not the hypotenuse) – in our case, side b. The three primary trigonometric ratios are: Sine (sin): sin(A) = Opposite / Hypotenuse = a/c Cosine (cos): cos(A) = Adjacent / Hypotenuse = b/c Tangent (tan): tan(A) = Opposite / Adjacent = a/b Mnemonic: A helpful mnemonic to remember these ratios is SOH CAH TOA: Sine = Opposite / Hypotenuse Cosine = Adjacent / Hypotenuse Tangent = Opposite / Adjacent 2.2 Finding Unknown Sides If we know the length of one side and the measure of one acute angle in a right-angled triangle, we can use trigonometric ratios to find the lengths of the other sides.

Example 1: In right-angled triangle PQR, where angle Q = 90°, angle P = 30°, and side QR = 5 cm, find the length of side PR (the hypotenuse).

Step 1: Identify the known and unknown: Angle P = 30° (reference angle) QR = 5 cm (opposite side to angle P) PR = ? (hypotenuse)

Step 2: Choose the appropriate trigonometric ratio: Since we know the opposite side and want to find the hypotenuse, we use the sine ratio: sin(P) = Opposite / Hypotenuse Step 3: Substitute the values: sin(30°) = 5 / PR Step 4: Solve for the unknown: PR = 5 / sin(30°) sin(30°) = 0.5 (This is a standard trigonometric value you should memorize or find on your calculator) PR = 5 / 0.5 PR = 10 cm Therefore, the length of side PR is 10 cm.

Example 2: In right-angled triangle XYZ, where angle Y = 90°, angle X = 60°, and side XY = 8 cm, find the length of side YZ (opposite to angle X).

Step 1: Identify the known and unknown: Angle X = 60° XY = 8 cm (adjacent side to angle X) YZ = ? (opposite side to angle X)

Step 2: Choose the appropriate trigonometric ratio: Since we know the adjacent side and want to find the opposite side, we use the tangent ratio: tan(X) = Opposite / Adjacent Step 3: Substitute the values: tan(60°) = YZ / 8 Step 4: Solve for the unknown: YZ = 8 tan(60°) tan(60°) ≈ 1.732 (Use your calculator to find this value) YZ ≈ 8 1.732 YZ ≈ 13.86 cm Therefore, the length of side YZ is approximately 13.86 cm. 2.3 Finding Unknown Angles If we know the lengths of two sides in a right-angled triangle, we can use trigonometric ratios to find the measure of the acute angles. We use the inverse trigonometric functions for this: arcsin (sin⁻¹), arccos (cos⁻¹), and arctan (tan⁻¹).

Example 3: In right-angled triangle KLM, where angle L = 90°, side KL = 6 cm, and side KM = 12 cm, find the measure of angle

K. Step 1: Identify the known and unknown: KL = 6 cm (adjacent side to angle K) KM = 12 cm (hypotenuse) Angle K = ?

Step 2: Choose the appropriate trigonometric ratio: Since we know the adjacent and hypotenuse, we use the cosine ratio: cos(K) = Adjacent / Hypotenuse Step 3: Substitute the values: cos(K) = 6 / 12 cos(K) = 0.5 Step 4: Solve for the unknown angle using the inverse cosine function: K = cos⁻¹(0.5) K = 60° (Use your calculator to find the inverse cosine) Therefore, the measure of angle K is 60°. 2.4 Angles of Elevation and Depression Angle of Elevation: The angle formed between the horizontal line and the line of sight when an observer looks up at an object.

Angle of Depression: The angle formed between the horizontal line and the line of sight when an observer looks down at an object.

Example 4: From the top of a building 50 meters high, the angle of depression to a car parked on the ground is 30°. Find the distance of the car from the base of the building.

Step 1: Draw a diagram. This is crucial for visualizing the problem.

Step 2: Identify the known and unknown. Height of building = 50 meters (opposite side to the angle of depression) Angle of depression = 30° Distance of car from the base of the building = ? (adjacent side to the angle of depression)

Step 3: Recognize that the angle of depression is equal to the angle of elevation from the car to the top of the building. This allows us to work with a standard right-angled triangle setup.

Step 4: Choose the appropriate trigonometric ratio: We know the opposite side and want to find the adjacent side, so we use the tangent ratio: tan(30°) = Opposite / Adjacent Step 5: Substitute the values: tan(30°) = 50 / Distance Step 6: Solve for the unknown: Distance = 50 / tan(30°) tan(30°) ≈ 0.577 (Use your calculator) Distance ≈ 50 / 0.577 Distance ≈ 86.65 meters Therefore, the car is approximately 86.65 meters away from the base of the building. Guided Practice (With Solutions)

Question 1: In right-angled triangle DEF, where angle E = 90°, angle D = 45°, and side EF = 7 cm, find the length of side DE.