Lesson Notes By Weeks and Term v5 - Grade 9

Lesson Video

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Performance objectives

• Identify opposite, adjacent, and hypotenuse sides.
• Define and apply sine, cosine, and tangent ratios.
• Calculate the length of an unknown side in a right-angled triangle.
• Use a calculator to find sin, cos, and tan of angles.

Lesson summary

This week, we delve into the exciting world of Measurement and Trigonometry. Trigonometry is much more than just abstract maths; it's a powerful tool used in various professions from architecture and engineering to surveying and navigation. Understanding trigonometry helps us solve real-world problems involving angles and distances, like calculating the height of a building or the distance across a valley. In the South African context, it is crucial for infrastructure development, land surveying, and even understanding weather patterns and their impact on agriculture. This week, we'll concentrate on finding the sides of right-angled triangles using trigonometric ratios.

Lesson notes

Understanding the Right-Angled Triangle Trigonometry, as we're studying it now, focuses on right-angled triangles. A right-angled triangle has one angle that measures exactly 90 degrees. The side opposite the right angle is the hypotenuse. It's also the longest side of the triangle. The other two sides are named relative to the angle we're interested in (other than the 90-degree angle).

Opposite: The side opposite to the angle we are considering.

Adjacent: The side next to the angle we are considering (that is not the hypotenuse).

Hypotenuse: The side opposite the right angle (the longest side). It’s crucial to identify these correctly, as they form the basis of our trigonometric ratios. Imagine you're standing at a specific angle within the triangle; the "opposite" side is what you see directly across from you, while the "adjacent" is the side you're effectively standing on (excluding the hypotenuse, which is always constant).

Trigonometric Ratios: SOH CAH TOA The three primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). These ratios relate the angles of a right-angled triangle to the lengths of its sides.

They are defined as follows: Sine (sin) = Opposite / Hypotenuse (SOH) Cosine (cos) = Adjacent / Hypotenuse (CAH) Tangent (tan) = Opposite / Adjacent (TOA) A helpful mnemonic to remember these ratios is SOH CAH TOA. Using a Calculator Your scientific calculator is crucial. Make sure it is in "degree" mode (usually indicated by a "D" or "DEG" on the display). To find the sine, cosine, or tangent of an angle, simply enter the angle followed by the sin, cos, or tan button, respectively. For example, to find sin(30°), you would press "30", then "sin" (or "sin", then "30", depending on the calculator model).

Worked example

Example 1:

Consider a right-angled triangle ABC, where angle B is 90 degrees, angle A is 30 degrees, and the hypotenuse AC is 10 cm. Find the length of side BC (opposite to angle A).

Solution:

Identify the relevant ratio: We are given the hypotenuse and we need to find the opposite side. The ratio that relates opposite and hypotenuse is sine (SOH).

Write the equation: sin(A) = Opposite / Hypotenuse

Substitute the values: sin(30°) = BC / 10 cm

Solve for BC: BC = 10 cm * sin(30°)

Use a calculator: sin(30°) = 0.5

Calculate BC: BC = 10 cm * 0.5 = 5 cm

Therefore, the length of side BC is 5 cm.