Study Notes: Grade 10 Mathematics, Trigonometric Functions

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Grade 10 Mathematics: Trigonometric Functions

1. Topic Overview

Main Concept/Theme:
Trigonometric functions explore the relationships between the angles and sides of a triangle. In Grade 10, we delve into the fundamental trigonometric ratios—sine, cosine, and tangent—and their applications in solving problems related to angles and distances.

Key Learning Objectives:
– Understand and define the trigonometric ratios: sine, cosine, and tangent.
– Apply trigonometric ratios to right-angled triangles.
– Use trigonometric ratios to solve problems involving angles and distances.
– Understand the unit circle and how it relates to trigonometric functions.
– Use calculators to find the values of trigonometric functions.

2. Key Terms and Definitions

  • Angle: A measure of rotation between two intersecting lines or planes.
  • Right-Angled Triangle: A triangle that contains a 90-degree angle.
  • Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.
  • Opposite side: The side opposite the angle in question in a triangle.
  • Adjacent side: The side next to the angle in question in a triangle.
  • Sine (sin): A trigonometric function defined as the ratio of the opposite side to the hypotenuse.
  • Cosine (cos): A trigonometric function defined as the ratio of the adjacent side to the hypotenuse.
  • Tangent (tan): A trigonometric function defined as the ratio of the opposite side to the adjacent side.
  • Unit Circle: A circle with a radius of one, centered at the origin of a coordinate system.

3. Main Content Sections

3.1 Introduction to Trigonometric Ratios

In a right-angled triangle, the trigonometric ratios are defined as follows:
– Sine (sin θ) = Opposite / Hypotenuse
– Cosine (cos θ) = Adjacent / Hypotenuse
– Tangent (tan θ) = Opposite / Adjacent

Example Triangle

Consider a right-angled triangle with angle θ:

|\
o | \ h
| \
90°|___\ (θ)
a

where:
– o = Opposite side to θ
– a = Adjacent side to θ
– h = Hypotenuse

3.2 Using Trigonometric Ratios to Find Unknown Sides and Angles

To solve problems involving right-angled triangles:
1. Identify which sides of the triangle are known and which are unknown.
2. Choose the appropriate trigonometric ratio based on the known and unknown sides.
3. Set up the equation and solve for the unknown side or angle.

Example Problem:

Given a right-angled triangle where:
– The angle θ is 30 degrees.
– The length of the hypotenuse is 10 units.

Find the length of the opposite side.

Solution:
Using sine:
sin(30°) = Opposite / Hypotenuse
0.5 = Opposite / 10
Opposite = 0.5 × 10 = 5 units

3.3 The Unit Circle

The unit circle is a fundamental concept in trigonometry. It’s a circle with a radius of 1, centered at the origin (0,0) of a coordinate system. The coordinates of points on the unit circle correspond to the cosine and sine of the angles formed by the radius.

Key Points on the Unit Circle:

  • The point (1, 0) corresponds to 0° or 360° (cos(0°) = 1, sin(0°) = 0).
  • The point (0, 1) corresponds to 90° (cos(90°) = 0, sin(90°) = 1).
  • The point (-1, 0) corresponds to 180° (cos(180°) = -1, sin(180°) = 0).
  • The point (0, -1) corresponds to 270° (cos(270°) = 0, sin(270°) = -1).

3.4 Calculator Use

Modern calculators have functions for finding the sine, cosine, and tangent of angles. Ensure your calculator is in the correct mode (degree or radian) based on the angle measure you are working with.

4. Example Problems

Problem 1:

A ladder leans against a wall, forming a 60-degree angle with the ground. If the length of the ladder is 12 units, find the height at which the ladder touches the wall.

Solution:
Using sine:
sin(60°) = Opposite / Hypotenuse
√3/2 = Height / 12
Height = 12 × √3/2 = 6√3 ≈ 10.39 units

Problem 2:

Find the cosine of an angle θ in a right-angled triangle where the adjacent side is 8 units, and the hypotenuse is 10 units.

Solution:
cos(θ) = Adjacent / Hypotenuse
cos(θ) = 8 / 10
cos(θ) = 0.8

5. Summary

  • Trigonometric functions relate the angles and sides of right-angled triangles.
  • Key trigonometric ratios are sine (sin), cosine (cos), and tangent (tan).
  • The unit circle helps visualize angles and their sine and cosine values.
  • Use calculators to find trigonometric values, ensuring correct mode settings.

6. Self-Assessment Questions

Multiple Choice:

  1. What is the sine of a 45° angle?
    a) 0.5
    b) √2/2
    c) √3/2
    d) 1
  2. What is the cosine of a 90° angle?
    a) 0
    b) 0.5
    c) 1
    d) -1

Open-Ended:

  1. Explain how to use the tangent ratio to find the height of a tree if you know the distance from the tree and the angle of elevation.
  2. Show how to find the sine and cosine of 30° using the unit circle.

7. Connections to Other Topics/Subjects

  • Geometry: Understanding the properties and measurements of triangles.
  • Physics: Applying trigonometry to solve problems involving forces and motion.
  • Geography: Using trigonometry in navigation and map reading.
  • Real-World Applications: Architecture, engineering, astronomy, and various fields requiring measurement and precision.

Remember, practicing problems and visualizing concepts through diagrams or real-life examples can significantly enhance your understanding of trigonometric functions. Happy studying!