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.
• Find unknown side lengths in right-angled triangles.
• Find unknown angles in right-angled triangles.
• Solve practical problems using angles of elevation and depression.

Lesson summary

Trigonometry, derived from Greek words meaning "triangle measurement," is a fundamental branch of mathematics that explores the relationships between the sides and angles of triangles. In Grade 10, we build upon the basics to explore trigonometric ratios in right-angled triangles and their applications. This knowledge is not just abstract theory; it has practical applications in fields like surveying, navigation, architecture, engineering, and even game development.

Lesson notes

2.1 Trigonometric Ratios: Sine, Cosine, and Tangent In a right-angled triangle, the sides are named relative to a specific acute angle (an angle less than 90 degrees).

Hypotenuse: The side opposite the right angle (the longest side).

Opposite: The side opposite the angle in question.

Adjacent: The side next to the angle in question (not the hypotenuse). The three primary trigonometric ratios are defined as follows: Sine (sin): sin(θ) = Opposite / Hypotenuse Cosine (cos): cos(θ) = Adjacent / Hypotenuse Tangent (tan): tan(θ) = Opposite / Adjacent A helpful mnemonic to remember these is SOH CAH TOA: Sine = Opposite / Hypotenuse Cosine = Adjacent / Hypotenuse Tangent = Opposite / Adjacent 2.2 Finding Unknown Sides If we know one side and one acute angle in a right-angled triangle, we can use trigonometric ratios to find the other sides.

Example 1: A right-angled triangle has an angle of 30 degrees. The hypotenuse is 10 cm long. Find the length of the side opposite the 30-degree angle.

Step 1: Identify what we know. Angle (θ) = 30° Hypotenuse = 10 cm We want to find the Opposite side.

Step 2: Choose the correct trigonometric ratio. Since we know the Hypotenuse and want to find the Opposite, we use Sine (SOH). sin(θ) = Opposite / Hypotenuse Step 3: Substitute the known values. sin(30°) = Opposite / 10 Step 4: Solve for the unknown. Opposite = 10 sin(30°) Opposite = 10 0.5 (sin(30°) = 0.5) Opposite = 5 cm Therefore, the length of the side opposite the 30-degree angle is 5 cm.

Example 2: 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 degrees. How far is the base of the ladder from the wall?

Step 1: Identify what we know. Angle (θ) = 60° Hypotenuse = 5 m (the ladder) We want to find the Adjacent side (distance from the wall).

Step 2: Choose the correct trigonometric ratio. Since we know the Hypotenuse and want to find the Adjacent, we use Cosine (CAH). cos(θ) = Adjacent / Hypotenuse Step 3: Substitute the known values. cos(60°) = Adjacent / 5 Step 4: Solve for the unknown. Adjacent = 5 cos(60°) Adjacent = 5 0.5 (cos(60°) = 0.5) Adjacent = 2.5 m Therefore, the base of the ladder is 2.5 meters from the wall. 2.3 Finding Unknown Angles If we know two sides of a right-angled triangle, we can use inverse trigonometric functions (arcsin, arccos, arctan) to find the unknown angles. These are often denoted as sin -1 , cos -1 , and tan -1 on your calculator (usually accessed with the "shift" or "2nd" button).

Example 3: In a right-angled triangle, the opposite side is 4 cm and the adjacent side is 3 cm. Find the angle.

Step 1: Identify what we know. Opposite = 4 cm Adjacent = 3 cm We want to find the angle (θ).

Step 2: Choose the correct trigonometric ratio. Since we know the Opposite and Adjacent, we use Tangent (TOA). tan(θ) = Opposite / Adjacent Step 3: Substitute the known values. tan(θ) = 4 / 3 Step 4: Solve for the unknown. θ = tan -1 (4/3) θ ≈ 53.13° (using a calculator) Therefore, the angle is approximately 53.13 degrees. 2.4 Angles of Elevation and Depression Angle of Elevation: The angle between the horizontal line of sight and an object above the horizontal. Imagine you are standing on the ground looking up at the top of a tree. The angle between your horizontal gaze and your gaze upwards to the treetop is the angle of elevation.

Angle of Depression: The angle between the horizontal line of sight and an object below the horizontal. Imagine you are standing on top of a building looking down at a car. The angle between your horizontal gaze and your gaze downwards to the car is the angle of depression. Crucially, the angle of elevation from point A to point B is equal to the angle of depression from point B to point A (assuming A and B are in the same vertical plane).

Example 4: From the top of a 50-meter cliff, the angle of depression to a boat is 35 degrees. How far is the boat from the base of the cliff?

Step 1: Draw a diagram. This is very important for word problems.

Step 2: Identify what we know. Height of cliff = 50 m (this is the opposite side to the angle) Angle of depression = 35° (This is equal to the angle of elevation from the boat to the top of the cliff) We want to find the distance from the boat to the base of the cliff (adjacent side).

Step 3: Choose the correct trigonometric ratio. Since we know the Opposite and want to find the Adjacent, we use Tangent (TOA). tan(θ) = Opposite / Adjacent Step 4: Substitute the known values. tan(35°) = 50 / Adjacent Step 5: Solve for the unknown. Adjacent = 50 / tan(35°) Adjacent ≈ 71.4 m (using a calculator) Therefore, the boat is approximately 71.4 meters from the base of the cliff. Guided Practice (With Solutions)

Question 1: In a right-angled triangle ABC, angle A = 90°, angle B = 60°, and AB = 8 cm. Find the length of BC (the hypotenuse).

Solution: Step 1: Draw a diagram (very helpful!).

Step 2: Identify known and unknown.