EDEXCEL GCSE FOUNDATION MATHS - What are Exact Trig Values

Edexcel GCSE Foundation Maths: What are Exact Trig Values?

Introduction

Exact trig values are the precise values of sine, cosine, and tangent for certain angles, often expressed as fractions or roots. They are fundamental in trigonometry and crucial for solving many problems.

Key Angles and Their Exact Trig Values

The angles you need to know for GCSE are:

• 0°
• 30°
• 45°
• 60°
• 90°

Here's a table summarizing their exact trig values:

Understanding the Values

You can visualize these values using a unit circle. The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane.

1. Angles: The angles are measured counterclockwise from the positive x-axis.
2. Coordinates: The coordinates of the points where the angle lines intersect the unit circle represent the cosine and sine of that angle.
   • x-coordinate = cosine
   • y-coordinate = sine
3. Tangent: The tangent of an angle is the sine divided by the cosine.

Example: For 30°, the coordinates are (?3/2, 1/2). Therefore:

• sin 30° = 1/2
• cos 30° = ?3/2
• tan 30° = (1/2) / (?3/2) = 1/?3

Memorizing the Values

You don't have to memorize all the values from scratch. There are helpful patterns and tricks:

• SOH CAH TOA: This mnemonic helps you remember the relationships between sine, cosine, tangent, opposite, adjacent, and hypotenuse sides of a right triangle.
• Special Triangles: 30-60-90 and 45-45-90 triangles have specific side ratios that help you derive the exact trig values.
• Symmetry: The unit circle exhibits symmetry, which can help you deduce the values for angles in different quadrants.

Using Exact Trig Values

Exact trig values are used in many areas of mathematics and physics, including:

• Solving trigonometric equations: Finding solutions to equations involving sine, cosine, or tangent.
• Calculating lengths and angles in right triangles: Using SOH CAH TOA to determine missing sides or angles.
• Deriving other trigonometric identities: Establishing relationships between different trigonometric functions.

Practice

To solidify your understanding, practice:

• Identifying exact trig values for different angles.
• Using exact trig values to solve trigonometric equations.
• Applying exact trig values to real-world problems involving triangles.

Conclusion

Understanding exact trig values is fundamental for success in GCSE Maths. By visualizing the unit circle, learning the values for key angles, and practicing problem-solving, you can master this crucial concept and excel in trigonometry.