Mathematics
Grade 11
15 min
Trigonometric identities: Set 2
Trigonometric identities: Set 2
What you'll learn
- Identify the nearest ten to a given two-digit number in at least 8 out of 10 attempts.
- Solve rounding puzzles involving finding numbers that round to a specific ten, providing at least 3 correct solutions per puzzle.
- Explain the rule for rounding to the nearest ten (5 or more, round up; less than 5, round down) using their own words, with at least 80% accuracy.
- Apply rounding skills to estimate sums and differences in word problems, achieving at least 70% accuracy in problem-solving.
Tutorial Preview
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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...
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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...
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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.
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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
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What grade level is "Trigonometric identities: Set 2"?
Trigonometric identities: Set 2 is a Grade 11 Mathematics lesson on ExcelOS.
What will I learn in Trigonometric identities: Set 2?
You'll be able to: Identify the nearest ten to a given two-digit number in at least 8 out of 10 attempts; Solve rounding puzzles involving finding numbers that round to a specific ten, providing at least 3 correct solutions per puzzle; Explain the….
Is "Trigonometric identities: Set 2" free to practice?
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This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.