Lesson Notes By Weeks and Term v5 - Grade 9

Lesson Video

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Performance objectives

• Define and apply sine, cosine, and tangent ratios.
• Calculate unknown side lengths in right-angled triangles.
• Calculate unknown angle sizes in right-angled triangles.
• Solve word problems with angles of elevation and depression.

Lesson summary

This week, we delve into the fascinating world of trigonometry, building upon our knowledge of measurement. Trigonometry is the study of relationships between the sides and angles of triangles. It’s not just an abstract mathematical concept; it has countless real-world applications, from surveying land to designing buildings and even calculating distances using satellite navigation. Think about cell phone towers – their optimal placement relies heavily on trigonometric principles. In South Africa, understanding land surveying and construction is crucial for infrastructure development, resource management, and community planning.

Lesson notes

Trigonometry focuses on the relationships between the angles and sides of right-angled triangles. Remember that a right-angled triangle has one angle of 90 degrees. The side opposite the right angle is called the hypotenuse.

Trigonometric Ratios: For an acute angle (less than 90°) in a right-angled triangle, we define three primary trigonometric ratios: Sine (sin): The sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. sin(angle) = Opposite / Hypotenuse Cosine (cos): The cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. cos(angle) = Adjacent / Hypotenuse Tangent (tan): The tangent of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. tan(angle) = Opposite / Adjacent Mnemonic: A helpful mnemonic to remember these ratios is SOH CAH TOA: SOH: Sine = Opposite / Hypotenuse CAH: Cosine = Adjacent / Hypotenuse TOA: Tangent = Opposite / Adjacent Angles of Elevation and Depression: Angle of Elevation: The angle of elevation is the angle formed between the horizontal line of sight and the line of sight upwards to an object. Imagine looking up at the top of a building.

Angle of Depression: The angle of depression is the angle formed between the horizontal line of sight and the line of sight downwards to an object. Imagine looking down from the top of a cliff at a boat in the sea. Important

Note: 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).

Using a Calculator: Make sure your calculator is in "degree" mode (DEG). Most calculators have dedicated buttons for sin, cos, and tan. To find the angle when you know the ratio (e.g., sin(x) = 0.5), you'll use the inverse trigonometric functions: sin -1 , cos -1 , or tan -1 (often labeled as asin, acos, or atan on your calculator). These are often accessed by pressing the "shift" or "2nd" function key.

Worked example

Example 1: Finding the length of a side.

A ladder leans against a wall, forming a right-angled triangle. The ladder is 5 meters long, and the angle between the ladder and the ground is 60°. How far is the base of the ladder from the wall?

Draw a diagram: Sketch a right-angled triangle, labeling the hypotenuse (ladder) as 5m, the angle as 60°, and the distance from the wall (adjacent side) as 'x'.

Identify the relevant ratio: We know the hypotenuse and want to find the adjacent side.

Therefore, we use cosine.

Write the equation: cos(60°) = Adjacent / Hypotenuse => cos(60°) = x / 5

Solve for x: x = 5 * cos(60°)

Calculate: Using a calculator, cos(60°) = 0.5, so x = 5 * 0.5 = 2.5 meters.

Example 2: Finding the size of an angle.

A telephone pole is supported by a wire. The wire is attached to the ground 8 meters away from the base of the pole. The wire is 10 meters long. What is the angle of elevation of the wire?

Draw a diagram: Sketch a right-angled triangle, labeling the adjacent side as 8m, the hypotenuse as 10m, and the angle of elevation as 'θ'.

Identify the relevant ratio: We know the adjacent side and the hypotenuse.

Therefore, we use cosine.

Write the equation: cos(θ) = Adjacent / Hypotenuse => cos(θ) = 8 / 10 = 0.8

Solve for θ: θ = cos -1 (0.8)

Calculate: Using a calculator, θ ≈ 36.87 degrees.