Lesson Notes By Weeks and Term v5 - Grade 10
Trigonometric functions – Week 7 focus
Download the Lessonotes Mobile South Africa app for faster lesson access on Android and iPhone.
Trigonometric functions – Week 7 focus
Download the Lessonotes Mobile South Africa app for faster lesson access on Android and iPhone.
Subject: Mathematics
Class: Grade 10
Term: 2nd Term
Week: 7
Theme: General lesson support
This page supports the lesson note with a companion video and a short classroom-ready summary.
For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.
This week, we delve into the exciting world of trigonometric functions. Trigonometry is a fundamental branch of mathematics that explores the relationships between angles and sides of triangles. It's not just abstract theory; trigonometry is used every day in various fields, from architecture and engineering to navigation and surveying. For South African learners, understanding trigonometry can unlock opportunities in careers related to construction, renewable energy (solar panel placement relies on angles), and even wildlife management (calculating distances and heights). It is crucial for further studies in mathematics and physics.
2.1 The Right-Angled Triangle Trigonometry, at its core, deals with right-angled triangles. A right-angled triangle has one angle that measures exactly 90 degrees. The side opposite the right angle is called the hypotenuse (always the longest side). The other two sides are named relative to a specific acute angle (less than 90 degrees) that we are considering: Opposite: The side opposite to the angle.
Adjacent: The side adjacent to the angle (next to the angle, and NOT the hypotenuse).
Diagram: ``` Hypotenuse /| / | Opposite / | /___| Angle Adjacent ``` 2.2 Trigonometric Ratios The three fundamental trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). These ratios relate the sides of a right-angled triangle to its angles.
They are defined as follows: Sine (sin): sin(angle) = Opposite / Hypotenuse Cosine (cos): cos(angle) = Adjacent / Hypotenuse Tangent (tan): tan(angle) = Opposite / Adjacent Mnemonic: A helpful way to remember these ratios is SOH CAH TOA: Sine = Opposite / Hypotenuse Cosine = Adjacent / Hypotenuse Tangent = Opposite / Adjacent 2.3 Finding Trigonometric Ratios with a Calculator Your scientific calculator has buttons for sin, cos, and tan. To find the sine, cosine, or tangent of an angle, simply enter the angle's value and press the corresponding button. Make sure your calculator is in "degree" mode (usually indicated by "DEG" or "D"). If it's in radian mode ("RAD" or "R"), the results will be incorrect. 2.4 Solving for 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: In a right-angled triangle, angle A = 30°, the hypotenuse = 10cm. Find the length of the side opposite angle
A. Solution: Identify what we know: Angle A = 30° Hypotenuse = 10 cm We want to find the Opposite side. Choose the appropriate trigonometric ratio: Since we know the Hypotenuse and want to find the Opposite, we use the sine ratio: sin(angle) = Opposite / Hypotenuse Substitute the known values: sin(30°) = Opposite / 10 Solve for the unknown: Opposite = 10 sin(30°) Opposite = 10 0.5 (sin(30°) = 0.5) Opposite = 5 cm Example 2: 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?
Solution: Draw a diagram: This helps visualize the problem. The ladder is the hypotenuse, the wall is the opposite side, and the distance from the wall to the base of the ladder is the adjacent side.
Identify what we know: Angle = 60° Adjacent = 2 meters We want to find the Opposite side. Choose the appropriate trigonometric ratio: Since we know the Adjacent and want to find the Opposite, we use the tangent ratio: tan(angle) = Opposite / Adjacent Substitute the known values: tan(60°) = Opposite / 2 Solve for the unknown: Opposite = 2 tan(60°) Opposite = 2 1.732 (approximately) Opposite = 3.464 meters (approximately) 2.5 Solving for Unknown Angles If we know the lengths of two sides of a right-angled triangle, we can use inverse trigonometric functions to find the angles.
The inverse trigonometric functions are: arcsin (or sin -1 ) - Used to find an angle when we know the sine ratio. arccos (or cos -1 ) - Used to find an angle when we know the cosine ratio. arctan (or tan -1 ) - Used to find an angle when we know the tangent ratio.
Example 3: In a right-angled triangle, the opposite side is 4 cm, and the adjacent side is 3 cm. Find the angle opposite the 4 cm side.
Solution: Identify what we know: Opposite = 4 cm Adjacent = 3 cm We want to find the angle. Choose the appropriate trigonometric ratio: Since we know the Opposite and Adjacent, we use the tangent ratio: tan(angle) = Opposite / Adjacent Substitute the known values: tan(angle) = 4 / 3 Solve for the unknown angle using the inverse tangent function (arctan or tan -1 ): angle = tan -1 (4/3) angle ≈ 53.13° (using a calculator)
Example 4: A cellphone tower is supported by a cable that runs from the top of the tower to a point on the ground 15m from the base of the tower. If the cable is 25m long, what is the angle of elevation of the cable (the angle it makes with the ground)?
Solution: Draw a diagram Identify what we know: Adjacent = 15m Hypotenuse = 25m We want to find the angle of elevation Choose the appropriate trigonometric ratio: Since we know the Adjacent and the Hypotenuse, we use the cosine ratio: cos(angle) = Adjacent / Hypotenuse Substitute the known values: cos(angle) = 15 / 25 = 3/5 Solve for the unknown angle using the inverse cosine function (arccos or cos -1 ): angle = cos -1 (3/5) angle ≈ 53.13° (using a calculator) Guided Practice (With Solutions)
Question 1: In right-angled triangle ABC, with angle B = 90°, AB = 8 cm, and BC = 6 cm. Find the values of sin(A), cos(A), and tan(A).