Mathematics
Grade 11
15 min
Trigonometric identities: Set 2
Trigonometric identities: Set 2
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Derive the sum and difference formulas for sine, cosine, and tangent.
Apply sum and difference formulas to find exact trigonometric values for non-standard angles (e.g., 15°, 75°).
Simplify complex trigonometric expressions using sum and difference identities.
Derive the double-angle formulas for sine, cosine, and tangent from the sum formulas.
Apply double-angle formulas to solve trigonometric equations and simplify expressions.
Prove more complex trigonometric identities using the sum, difference, and double-angle formulas.
Ever wondered how to find the exact value of sin(75°) without a calculator? 🤔 It's not a special angle, but it's the sum of two special angles!
This tutorial moves beyond basic identities to explore the powerful sum, diff...
2
Key Concepts & Vocabulary
TermDefinitionExample
Sum IdentityAn identity that expresses a trigonometric function of a sum of two angles (A + B) in terms of trigonometric functions of the individual angles A and B.cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
Difference IdentityAn identity that expresses a trigonometric function of a difference of two angles (A - B) in terms of trigonometric functions of the individual angles A and B.sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
Double-Angle IdentityA special case of a sum identity where the two angles are equal (A + A = 2A), expressing a trigonometric function of 2A in terms of functions of A.sin(2A) = 2sin(A)cos(A)
Exact ValueThe value of a trigonometric function expressed as a fraction or using radicals, not a decimal approximation.The exact value of cos(30°) is \frac{\sqr...
3
Core Formulas
Sum and Difference Formulas
sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B) \\ cos(A ± B) = cos(A)cos(B) ∓ sin(A)sin(B) \\ tan(A ± B) = \frac{tan(A) ± tan(B)}{1 ∓ tan(A)tan(B)}
Use these to find exact values of combined angles (like 75° = 45° + 30°) or to expand/condense trigonometric expressions. Note the sign change in the cosine formula.
Double-Angle Formulas
sin(2A) = 2sin(A)cos(A) \\ cos(2A) = cos^2(A) - sin^2(A) = 2cos^2(A) - 1 = 1 - 2sin^2(A) \\ tan(2A) = \frac{2tan(A)}{1 - tan^2(A)}
Use these when you need to relate a trigonometric function of an angle 2A to functions of the angle A. The three forms for cos(2A) are interchangeable and chosen based on the problem's needs.
4 more steps in this tutorial
Sign up free to access the complete tutorial with worked examples and practice.
Sign Up Free to ContinueSample Practice Questions
Challenging
Which of the following is a valid trigonometric identity derived from the sum and difference formulas?
A.sin(x+y)sin(x-y) = cos²(y) - cos²(x)
B.sin(x+y)sin(x-y) = sin²(x) + sin²(y)
C.sin(x+y)sin(x-y) = sin²(x) - sin²(y)
D.sin(x+y)sin(x-y) = 2sin(x)cos(y)
Challenging
Find all solutions to the equation cos(2x) + 3sin(x) - 2 = 0 in the interval [0, 2π).
A.{π/3, 2π/3}
B.{π/6, 5π/6, π/2}
C.{π/6, 5π/6}
D.{π/2, 7π/6, 11π/6}
Challenging
If A and B are acute angles such that A + B = π/4, what is the value of the expression (1 + tan(A))(1 + tan(B))?
A.1
B.√2
C.1 + √2
D.2
Want to practice and check your answers?
Sign up to access all questions with instant feedback, explanations, and progress tracking.
Start Practicing Free