Lesson Notes By Weeks and Term v5 - Grade 9
Measurement and trigonometry (Grade 9) – Week 10 focus
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Measurement and trigonometry (Grade 9) – Week 10 focus
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Subject: Mathematics
Class: Grade 9
Term: 3rd Term
Week: 10
Theme: General lesson support
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This week, we delve into the fascinating world of trigonometry, specifically focusing on calculating trigonometric ratios (sine, cosine, and tangent) for acute angles and applying them to solve problems involving right-angled triangles. Understanding trigonometry is crucial because it helps us determine unknown sides and angles in right-angled triangles, which has practical applications in various fields like surveying, navigation, architecture, and engineering. Imagine designing a ramp for wheelchair access or calculating the height of a cellphone tower - trigonometry is essential for these tasks!
2.1 Right-Angled Triangles: A Quick Recap A right-angled triangle is a triangle containing one angle of 90 degrees. The side opposite the right angle is called the hypotenuse (the longest side). The other two sides are referred to in relation to a specific acute angle (an angle less than 90 degrees) within the triangle.
Opposite Side: The side directly across from the angle we are considering.
Adjacent Side: The side next to the angle we are considering (that is not the hypotenuse). 2.2 Trigonometric Ratios: Defining Sine, Cosine, and Tangent Trigonometric ratios are ratios of the lengths of the sides of a right-angled triangle.
The three basic trigonometric ratios are: Sine (sin): The ratio of the length of the opposite side to the length of the hypotenuse. sin(angle) = Opposite / Hypotenuse Cosine (cos): The ratio of the length of the adjacent side to the length of the hypotenuse. cos(angle) = Adjacent / Hypotenuse Tangent (tan): The ratio of the length of the opposite side to the length of the adjacent side. tan(angle) = Opposite / Adjacent A helpful mnemonic to remember these ratios is SOH CAH TOA: Sin = Opposite / Hypotenuse Cosine = Adjacent / Hypotenuse Tangent = Opposite / Adjacent 2.3 Finding Trigonometric Ratios for a Given Angle Example 1: Consider a right-angled triangle ABC, where angle B is 90 degrees. Side AB = 4cm, side BC = 3cm, and side AC = 5cm. Find sin(A), cos(A), and tan(A).
Step 1: Identify the sides relative to angle
A. Opposite (to angle A) = BC = 3cm Adjacent (to angle A) = AB = 4cm Hypotenuse = AC = 5cm Step 2: Apply the trigonometric ratios. sin(A) = Opposite / Hypotenuse = 3/5 = 0.6 cos(A) = Adjacent / Hypotenuse = 4/5 = 0.8 tan(A) = Opposite / Adjacent = 3/4 = 0.75 Example 2: Consider a right-angled triangle PQR, where angle Q is 90 degrees. Side PQ = 12cm, and PR = 13cm. Calculate sin(P), cos(P) and tan(P).
Step 1: Find the length of QR using Pythagoras' theorem. PR 2 = PQ 2 + QR 2 13 2 = 12 2 + QR 2 169 = 144 + QR 2 QR 2 = 25 QR = 5cm Step 2: Identify the sides relative to angle
P. Opposite (to angle P) = QR = 5cm Adjacent (to angle P) = PQ = 12cm Hypotenuse = PR = 13cm Step 3: Apply the trigonometric ratios. sin(P) = Opposite / Hypotenuse = 5/13 ≈ 0.385 cos(P) = Adjacent / Hypotenuse = 12/13 ≈ 0.923 tan(P) = Opposite / Adjacent = 5/12 ≈ 0.417 2.4 Solving for Unknown Sides Using Trigonometric Ratios Example 3: In a right-angled triangle XYZ, angle Y is 90 degrees. Angle X = 30 degrees, and side XZ (hypotenuse) = 10m. Find the length of side YZ (opposite to angle X).
Step 1: Identify the knowns and unknowns. Angle X = 30 degrees Hypotenuse (XZ) = 10m Opposite (YZ) = ?
Step 2: Choose the appropriate trigonometric ratio. Since we have the hypotenuse and want to find the opposite side, we use sine: sin(angle) = Opposite / Hypotenuse Step 3: Substitute the known values and solve for the unknown. sin(30°) = YZ / 10 YZ = 10 sin(30°) YZ = 10 0.5 (sin(30°) = 0.5) YZ = 5m Example 4: In a right-angled triangle DEF, angle E is 90 degrees. Angle D = 60 degrees, and side DE (adjacent to angle D) = 7cm. Find the length of side EF (opposite to angle D).
Step 1: Identify the knowns and unknowns. Angle D = 60 degrees Adjacent (DE) = 7cm Opposite (EF) = ?
Step 2: Choose the appropriate trigonometric ratio. Since we have the adjacent and want to find the opposite side, we use tangent: tan(angle) = Opposite / Adjacent Step 3: Substitute the known values and solve for the unknown. tan(60°) = EF / 7 EF = 7 tan(60°) EF ≈ 7 1.732 (tan(60°) ≈ 1.732) EF ≈ 12.124cm 2.5 Solving for Unknown Angles Using Trigonometric Ratios When two sides of a right-angled triangle are known, we can use the inverse trigonometric functions to find an unknown angle. These are denoted as sin -1 (arcsin), cos -1 (arccos), and tan -1 (arctan). Most calculators have these functions (often accessed by pressing the "shift" or "2nd" button followed by the sin, cos, or tan button).
Example 5: In a right-angled triangle ABC, angle B is 90 degrees. Side AB = 8cm, and side BC = 6cm. Find the measure of angle
A. Step 1: Identify the sides relative to angle
A. Opposite (to angle A) = BC = 6cm Adjacent (to angle A) = AB = 8cm Step 2: Choose the appropriate trigonometric ratio. Since we have the opposite and adjacent sides, we use tangent: tan(A) = Opposite / Adjacent Step 3: Substitute the known values. tan(A) = 6/8 = 0.75 Step 4: Use the inverse tangent function (arctan or tan -1 ) to find angle
A. A = tan -1 (0.75) A ≈ 36.87 degrees Example 6: In a right-angled triangle PQR, angle Q is 90 degrees. Side PQ = 5cm, and side PR = 10cm. Find the measure of angle
P. Step 1: Identify the sides relative to angle
P. Adjacent (to angle P) = PQ = 5cm Hypotenuse = PR = 10cm Step 2: Choose the appropriate trigonometric ratio. Since we have the adjacent and hypotenuse, we use cosine: cos(P) = Adjacent / Hypotenuse Step 3: Substitute the known values. cos(P) = 5/10 = 0.5 Step 4: Use the inverse cosine function (arccos or cos -1 ) to find angle P.