EDEXCEL GCSE HIGHER MATHS - What are Exact Trigonometric Values and Identities

Edexcel GCSE Higher Maths: Exact Trigonometric Values and Identities

What are Exact Trigonometric Values?

Exact trigonometric values are the simplified values of trigonometric ratios (sine, cosine, tangent) for specific angles. These values are often expressed in terms of fractions and radicals (like square roots) rather than decimals.

Why are they important?

• Avoid rounding errors: Using exact values in calculations prevents rounding errors that occur when using decimal approximations.
• Simplify expressions: Exact values can be used to simplify trigonometric expressions and make them easier to work with.
• Solve trigonometric equations: They are crucial for finding the solutions of trigonometric equations, especially those that involve angles that are not multiples of 30° or 45°.

Special Angles and Their Exact Values:

How to remember these values:

• Use the SOH CAH TOA mnemonic to remember the relationships between sides and trigonometric ratios.
• Visualize the unit circle and use its symmetry to determine the signs of trigonometric values in different quadrants.
• Use the special triangles (30-60-90 and 45-45-90) to derive the values.

Trigonometric Identities:

Trigonometric identities are equations that are true for all values of the variable. They are used to:

• Simplify trigonometric expressions.
• Solve trigonometric equations.
• Prove other trigonometric identities.

Key Identities:

• Pythagorean Identity: sin²? + cos²? = 1
• Quotient Identity: tan? = sin? / cos?
• Reciprocal Identities:
  • csc? = 1 / sin?
  • sec? = 1 / cos?
  • cot? = 1 / tan?
• Angle Addition and Subtraction Formulas:
  • sin(? + ?) = sin?cos? + cos?sin?
  • sin(? - ?) = sin?cos? - cos?sin?
  • cos(? + ?) = cos?cos? - sin?sin?
  • cos(? - ?) = cos?cos? + sin?sin?
  • tan(? + ?) = (tan? + tan?) / (1 - tan?tan?)
  • tan(? - ?) = (tan? - tan?) / (1 + tan?tan?)
• Double Angle Formulas:
  • sin2? = 2sin?cos?
  • cos2? = cos²? - sin²? = 1 - 2sin²? = 2cos²? - 1
  • tan2? = 2tan? / (1 - tan²?)

Example:

Simplify the expression: sin²? + cos²? / tan?

Solution:

1. Apply the Pythagorean Identity: sin²? + cos²? = 1
2. Apply the Quotient Identity: tan? = sin? / cos?
3. Substitute the identities: 1 / (sin? / cos?)
4. Simplify: cos? / sin? = cot?

Practice:

• Calculate the exact value of sin 60° + cos 30°
• Prove the identity: tan²? + 1 = sec²?
• Solve the equation: sin²? - cos²? = 1/2

Remember to practice using the identities and exact values to build confidence and mastery in these concepts.