Gr12 Mathematics: Ch 4.1 Compound Angle Identities
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Questions and Answers

What is the formula for sin(α - β)?

  • cos α cos β + sin α sin β
  • cos α cos β - sin α sin β
  • sin α cos β - cos α sin β (correct)
  • sin α cos β + cos α sin β

    • What is the formula for cos(2α)?

  • cos^2 α + sin^2 α
  • 2 sin α cos α
  • 2 cos^2 α - 1 (correct)
  • cos^2 α - sin^2 α

    • What is the formula for sin(α + β)?

  • cos α cos β + sin α sin β
  • cos α cos β - sin α sin β
  • sin α cos β + cos α sin β (correct)
  • sin α cos β - cos α sin β

    • What is the formula for cos(α - β)?

    <p>cos α cos β - sin α sin β</p> Signup and view all the answers

    What is the formula for sin(2α)?

    <p>2 sin α cos α</p> Signup and view all the answers

    What is the formula for cos(α + β)?

    <p>cos α cos β - sin α sin β</p> Signup and view all the answers

    What is the formula for the cosine of a difference?

    <p>$\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$</p> Signup and view all the answers

    Which method is used to derive the formula for $\cos(\alpha + \beta)$?

    <p>Using the negative angle identity</p> Signup and view all the answers

    What is the formula for the sine of a sum?

    <p>$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$</p> Signup and view all the answers

    What is the formula for the cosine of a sum?

    <p>$\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$</p> Signup and view all the answers

    What is the purpose of deriving compound angle formulas?

    <p>To simplify trigonometric expressions involving sums and differences of angles</p> Signup and view all the answers

    What is the formula for the sine of a difference?

    <p>$\sin(\alpha - \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$</p> Signup and view all the answers

    Which of the following is not a method used to derive compound angle formulas?

    <p>Using the Pythagorean theorem</p> Signup and view all the answers

    What is the advantage of using compound angle formulas?

    <p>They can be used to simplify complex trigonometric expressions</p> Signup and view all the answers

    What is the correct application of the co-function identity in the derivation of the sine of a difference formula?

    <p>cos(90° - α) = sin α</p> Signup and view all the answers

    Which of the following is a correct application of the compound angle formula for sine?

    <p>sin(α + β) = sin α cos β + cos α sin β</p> Signup and view all the answers

    What is the result of applying the double angle formula for sine to sin(4α)?

    <p>2 sin(2α) cos(2α)</p> Signup and view all the answers

    Which of the following identities is used in the derivation of the cosine of a difference formula?

    <p>cos(α - β) = cos α cos β + sin α sin β</p> Signup and view all the answers

    What is the difference between the formulas for cos(α - β) and cos(α + β)?

    <p>A negative sign is present in the former and a positive sign in the latter.</p> Signup and view all the answers

    In deriving the formula for cos(α - β), which of the following is used?

    <p>The distance formula and cosine rule.</p> Signup and view all the answers

    What is the purpose of using co-functions in the derivation of compound angle formulas?

    <p>To simplify the expressions</p> Signup and view all the answers

    Which of the following formulas can be derived using the compound angle formula for cosine?

    <p>cos(2α) = cos^2 α - sin^2 α</p> Signup and view all the answers

    What is the purpose of rewriting α + β as α - (-β) in deriving the formula for cos(α + β)?

    <p>To apply the cosine difference formula.</p> Signup and view all the answers

    Which of the following is a consequence of the even-odd identities?

    <p>cos(-β) = cos(β) and sin(-β) = -sin(β).</p> Signup and view all the answers

    What is the relation between the formulas for sin(α - β) and sin(α + β)?

    <p>They differ only in the sign of the cosine terms.</p> Signup and view all the answers

    What is the underlying principle in deriving the formulas for compound angles?

    <p>The unit circle definition of trigonometric functions.</p> Signup and view all the answers

    Why are compound angle formulas useful?

    <p>They simplify trigonometric expressions involving sums and differences of angles.</p> Signup and view all the answers

    What is the significance of the method used to derive cos(α + β)?

    <p>It illustrates the importance of the even-odd identities.</p> Signup and view all the answers

    If sin(α) = 3/5 and cos(β) = 4/5, what is the value of sin(α - β)?

    <p>7/25</p> Signup and view all the answers

    If cos(α) = 2/3 and sin(β) = 1/3, what is the value of cos(α + β)?

    <p>4/9</p> Signup and view all the answers

    If sin(α) = 1/2 and cos(α) = √3/2, what is the value of sin(2α)?

    <p>√3/2</p> Signup and view all the answers

    If cos(α) = 3/5 and sin(α) = 4/5, what is the value of cos(2α)?

    <p>-7/25</p> Signup and view all the answers

    What is the value of sin(α) cos(β) + cos(α) sin(β) in terms of compound angle formulas?

    <p>sin(α + β)</p> Signup and view all the answers

    What is the value of cos(α) cos(β) - sin(α) sin(β) in terms of compound angle formulas?

    <p>cos(α + β)</p> Signup and view all the answers

    What is the key step in deriving the formula for cos(α + β) using the negative angle identity?

    <p>Rewriting the angle as a difference α - (-β)</p> Signup and view all the answers

    What is the purpose of using the distance formula and cosine rule in deriving the formula for cos(α - β)?

    <p>To equate two expressions for KL^2</p> Signup and view all the answers

    What is the common feature among the formulas for sin(α - β), sin(α + β), cos(α - β), and cos(α + β)?

    <p>They all involve the product of two trigonometric functions</p> Signup and view all the answers

    What is the reason for the difference in signs between the formulas for cos(α - β) and cos(α + β)?

    <p>Due to the difference in the direction of the angles</p> Signup and view all the answers

    What is the underlying principle behind the derivation of the formulas for compound angles?

    <p>The principle of equivalence of expressions</p> Signup and view all the answers

    What is the role of the negative angle identity in deriving the formula for cos(α + β)?

    <p>To apply the cosine difference formula</p> Signup and view all the answers

    What is the consequence of applying the even-odd identities in the derivation of the formulas for compound angles?

    <p>Simplification of the expressions</p> Signup and view all the answers

    What is the key idea behind the derivation of the formulas for cos(α - β) and cos(α + β)?

    <p>Equating two expressions for KL^2</p> Signup and view all the answers

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