Lesson Notes By Weeks and Term v5 - Grade 9

Lesson Video

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Performance objectives

• Identify hypotenuse, opposite, and adjacent sides.
• Define sine, cosine, and tangent (SOH CAH TOA).
• Use a calculator to find trig ratios and angles.
• Solve for unknown sides in right-angled triangles.
• Solve for unknown angles in right-angled triangles.

Lesson summary

This week, we delve into the fascinating world of measurement and trigonometry, building upon our previous understanding of geometric concepts. Trigonometry, derived from Greek words meaning "triangle measurement," is a branch of mathematics that explores the relationships between angles and sides of triangles. While it might seem abstract now, trigonometry is fundamental to many real-world applications, from surveying land for development in our communities to calculating the height of buildings or mountains, even navigating ships and airplanes.

Lesson notes

Right-Angled Triangles: The Foundation Trigonometry, as we're learning it this week, focuses on right-angled triangles. A right-angled triangle is a triangle that has one angle measuring exactly 90 degrees. The side opposite the right angle is always the longest side and is called the hypotenuse. The other two sides are called the opposite and the adjacent, but their names depend on which of the other two angles you are considering. Imagine a right-angled triangle ABC, with the right angle at

B. Hypotenuse: The side opposite the right angle (AC) is the hypotenuse. It's always the same, regardless of which angle we are focusing on. Now, let's consider angle A: Opposite: The side opposite to angle A (BC) is called the opposite side.

Adjacent: The side next to angle A (AB), that is not the hypotenuse, is called the adjacent side.

If we switch our focus to angle C: Opposite: The side opposite to angle C (AB) becomes the opposite side.

Adjacent: The side next to angle C (BC), that is not the hypotenuse, becomes the adjacent side.

Important: The hypotenuse never changes; it's always opposite the right angle. The opposite and adjacent sides change depending on which acute angle you are referring to.

Trigonometric Ratios: Sine, Cosine, and Tangent These ratios provide a link between the angles and side lengths of a right-angled triangle.

They are defined as follows: Sine (sin): The sine of an angle is the ratio of the length of the opposite side 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 adjacent side 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 opposite side to the length of the adjacent side. tan(angle) = Opposite / Adjacent A handy mnemonic to remember these ratios is SOH CAH TOA: SOH: Sine = Opposite / Hypotenuse CAH: Cosine = Adjacent / Hypotenuse TOA: Tangent = Opposite / Adjacent Using a Calculator to Find Trigonometric Ratios Your scientific calculator is an essential tool for trigonometry. Make sure your calculator is set to "degree" mode ("DEG" should be displayed). To find the sine, cosine, or tangent of an angle, simply: Press the "sin," "cos," or "tan" button. Enter the angle in degrees. Press the "=" button. For example, to find sin(30°): Press "sin" Enter "30" Press "=" The calculator should display 0.

5. To find an angle when you know the sine, cosine, or tangent value, you need to use the inverse trigonometric functions: sin -1 (arcsin), cos -1 (arccos), or tan -1 (arctan). These are usually accessed by pressing the "shift" or "2nd" button followed by the "sin," "cos," or "tan" button. For example, to find the angle whose sine is 0.5: Press "shift" or "2nd" Press "sin" (This will display sin -1 or arcsin) Enter "0.5" Press "=" The calculator should display

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0. Solving Problems with Trigonometric Ratios Let's look at some examples: Example 1: Finding a Missing Side A ladder leans against a wall, making an angle of 60° with the ground. The base of the ladder is 2 meters away from the wall. How high up the wall does the ladder reach?

Step 1: Draw a diagram. This is crucial for visualizing the problem. Draw a right-angled triangle where the wall is the vertical side, the ground is the horizontal side, and the ladder is the hypotenuse.

Step 2: Identify the known and unknown quantities.

Angle: 60° Adjacent side: 2 meters (distance from the wall to the base of the ladder)

Opposite side: x (height up the wall – this is what we want to find)

Step 3: Choose the appropriate trigonometric ratio. Since we know the adjacent side and want to find the opposite side, we use the tangent ratio: tan(angle) = Opposite / Adjacent Step 4: Substitute the known values and solve for the unknown. tan(60°) = x / 2 x = 2 tan(60°) Using a calculator, tan(60°) ≈ 1.732 x ≈ 2 1.732 x ≈ 3.464 meters Therefore, the ladder reaches approximately 3.464 meters up the wall.

Example 2: Finding a Missing Angle A ramp is 5 meters long and rises to a height of 1 meter. What is the angle of inclination of the ramp (the angle it makes with the ground)?

Step 1: Draw a diagram. Draw a right-angled triangle where the ramp is the hypotenuse, the height is the opposite side, and the horizontal distance is the adjacent side.

Step 2: Identify the known and unknown quantities.

Opposite side: 1 meter (height of the ramp)

Hypotenuse: 5 meters (length of the ramp)

Angle: θ (the angle of inclination – this is what we want to find)

Step 3: Choose the appropriate trigonometric ratio. Since we know the opposite side and the hypotenuse, we use the sine ratio: sin(angle) = Opposite / Hypotenuse Step 4: Substitute the known values and solve for the unknown. sin(θ) = 1 / 5 = 0.2 θ = sin -1 (0.2)

Step 5: Use the inverse sine function on your calculator. θ ≈ 11.54° Therefore, the angle of inclination of the ramp is approximately 11.54 degrees.