Gr12 Mathematics: Ch 4 Sum Trigonometry
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Questions and Answers

What is the formula for the sine of a difference?

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

What is the formula for the cosine of a double angle?

  • cos(2α) = 2 sin² α + 1
  • cos(2α) = cos² α - sin² α (correct)
  • cos(2α) = 1 - 2 cos² α
  • cos(2α) = 2 cos² α - 1 (correct)

What is the first step in solving a trigonometric equation?

  • Determine the reference angle
  • Determine where the function is positive or negative
  • Find angles within a specified interval
  • Simplify the equation using algebraic methods and trigonometric identities (correct)

What is the purpose of a CAST diagram?

<p>To determine where the function is positive or negative (C)</p> Signup and view all the answers

What is the formula for the sine of a sum?

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

What is the formula for the cosine of a sum?

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

What is the formula for the sine of a double angle?

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

What is the purpose of using a reference angle?

<p>To find the restricted values (A)</p> Signup and view all the answers

What is the formula for the cosine of a difference?

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

What is the general solution to a trigonometric equation?

<p>An infinite number of angles that satisfy the equation (B)</p> Signup and view all the answers

What is the formula for the cosine of a difference?

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

What is the formula for the sine of a sum?

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

What is the purpose of using the distance formula and cosine rule in deriving the formula for $\cos(\alpha - \beta)$?

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

What is the formula for the cosine of a sum?

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

What is the formula for the sine of a difference?

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

Why are the even-odd identities used in deriving the formula for $\cos(\alpha + \beta)$?

<p>To simplify the expression for $\cos(-\beta)$ (B)</p> Signup and view all the answers

What is the purpose of deriving the compound angle formulas?

<p>To express trigonometric functions of a sum or difference in terms of functions of the individual angles (C)</p> Signup and view all the answers

What is the formula to find the general solution for sin theta = x?

<p>theta = 180^degree - sin^(-1)x + k * 360^degree (B), theta = sin^(-1)x + k * 360^degree (C)</p> Signup and view all the answers

What is the formula for the area of a triangle using the area rule?

<p>All of the above (D)</p> Signup and view all the answers

When should you use the cosine rule?

<p>When no right angle is given, and either two sides and the included angle or three sides are given (A)</p> Signup and view all the answers

What is the formula for the height of a pole using the sine and tangent rules?

<p>h = (d sin alpha)/(sin beta) tan beta (D)</p> Signup and view all the answers

What is the formula for the height of a building using the sine rule?

<p>h = (b sin alpha sin theta)/(sin(beta + theta)) (D)</p> Signup and view all the answers

What is the formula for tan theta = x?

<p>theta = tan^(-1)x + k * 180^degree (B)</p> Signup and view all the answers

When should you use the sine rule?

<p>When no right angle is given, and two sides and an angle (not the included angle) are given (A)</p> Signup and view all the answers

What is the formula for the area of a triangle using the sine rule?

<p>Area = (1/2)ab sin C (D)</p> Signup and view all the answers

What is the formula for cos theta = x?

<p>theta = cos^(-1)x + k * 360^degree (A), theta = 360^degree - cos^(-1)x + k * 360^degree (C)</p> Signup and view all the answers

What is the formula for $\cos(\alpha - eta)$?

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

What is the formula for $\sin(\alpha + eta)$?

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

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

<p>Using the distance formula and cosine rule (D)</p> Signup and view all the answers

What is the purpose of using the distance formula and cosine rule in deriving the formula for $\cos(\alpha - eta)$?

<p>To derive the formula for $\cos(\alpha - eta)$ (A)</p> Signup and view all the answers

Why are the even-odd identities used in deriving the formula for $\cos(\alpha + eta)$?

<p>To rewrite the angle as a difference (A)</p> Signup and view all the answers

What is the formula for $\sin(\alpha - eta)$?

<p>$\sin \alpha \cos eta - \cos \alpha \sin eta$ (D)</p> Signup and view all the answers

What is the purpose of deriving the compound angle formulas?

<p>To extend trigonometric identities (B)</p> Signup and view all the answers

Which formula is used as a stepping stone to derive the formula for $\cos(\alpha + eta)$?

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

What is the formula for cos(α - β)?

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

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

<p>-1/9 (C)</p> Signup and view all the answers

What is the formula for sin(α + β)?

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

What is the purpose of using a CAST diagram in solving trigonometric equations?

<p>To determine where the function is positive or negative (C)</p> Signup and view all the answers

If cos(α) = 1/2, then what is the value of cos(2α)?

<p>3/4 (D)</p> Signup and view all the answers

What is the formula for cos(α + β)?

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

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

<p>sqrt(3)/2 (A)</p> Signup and view all the answers

What is the first step in solving a trigonometric equation?

<p>Simplify the equation using algebraic methods (A)</p> Signup and view all the answers

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

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

What is the general solution for the equation tan θ = x?

<p>θ = tan⁻¹x + k ⋅ 180°, where k is an integer (C)</p> Signup and view all the answers

What is the formula for the area of a triangle with sides a, b, and angle C?

<p>All of the above (D)</p> Signup and view all the answers

When should you use the sine rule?

<p>When no right angle is given, and either two sides and an angle or two angles and a side are given (C)</p> Signup and view all the answers

What is the formula for the height of a building using the sine rule?

<p>h = (b sin α sin θ) / (sin(β + θ)) (C)</p> Signup and view all the answers

What is the formula for the area of a triangle with sides a and b, and angle C?

<p>Area = (1/2)ab sin C (D)</p> Signup and view all the answers

When should you use the cosine rule?

<p>When no right angle is given, and three sides are given (B)</p> Signup and view all the answers

What is the formula for the height of a pole using the sine and tangent rules?

<p>h = (d sin α) / (sin β) tan β (D)</p> Signup and view all the answers

What is the general solution for the equation sin θ = x?

<p>θ = sin⁻¹x + k ⋅ 180° or θ = 180° - sin⁻¹x + k ⋅ 360° (B)</p> Signup and view all the answers

What is the formula for the area of a triangle with sides a, b, and c?

<p>Area = (1/2)ab sin C (D)</p> Signup and view all the answers

What is the general approach to solving three-dimensional problems?

<p>Draw a sketch, consider the given information, apply appropriate rules, and calculate the desired quantities (A)</p> Signup and view all the answers

What is the sine of a 30° angle minus a 45° angle?

<p>√2/2 - 1/2 (B)</p> Signup and view all the answers

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

<p>8√5/9 (A)</p> Signup and view all the answers

What is the cosine of a 30° angle plus a 45° angle?

<p>√2/2 + 1/2 (C)</p> Signup and view all the answers

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

<p>3/4 (A)</p> Signup and view all the answers

What is the sine of a 60° angle plus a 30° angle?

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

What is the cosine of a 45° angle minus a 30° angle?

<p>√2/2 - 1/2 (A)</p> Signup and view all the answers

If (\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta), then what is (\cos(\alpha + \beta))?

<p>\cos \alpha \cos \beta - \sin \alpha \sin \beta (C)</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 (A)</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(\alpha - \beta))?

<p>To simplify the expression ((\cos \alpha - \cos \beta)^2 + (\sin \alpha - \sin \beta)^2) (D)</p> Signup and view all the answers

What is the sine of a 75° angle?

<p>√2 - 1/2 (B)</p> Signup and view all the answers

What is the key step in deriving the formula for (\cos(\alpha + \beta))?

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

If (\sin \alpha = \frac{2}{3}) and (\cos \alpha = \frac{\sqrt{5}}{3}), then what is the value of (\cos(2\alpha))?

<p>\frac{-1}{9} (B)</p> Signup and view all the answers

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

<p>-4/9 (D)</p> Signup and view all the answers

What is the formula for (\sin(\alpha - \beta))?

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

If (\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta), then what is (\cos(\alpha - \beta))?

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

What is the key step in deriving the formula for (\sin(\alpha + \beta))?

<p>Applying the sine difference formula (A)</p> Signup and view all the answers

What is the purpose of deriving the compound angle formulas?

<p>To simplify trigonometric expressions (A)</p> Signup and view all the answers

What is the general solution to the equation sin(θ) = x in the interval [0°, 360°]?

<p>θ = sin⁻¹(x) + k × 360° (A)</p> Signup and view all the answers

For the equation cos(θ) = x, what is the value of θ in the interval [0°, 360°]?

<p>θ = cos⁻¹(x) or θ = 180° - cos⁻¹(x) (B)</p> Signup and view all the answers

What is the formula for the area of a triangle with sides a, b, and angle C?

<p>Area = (1/2)ab sin(C) (C)</p> Signup and view all the answers

When should you use the cosine rule in solving triangular problems?

<p>When no right angle is given, and either two sides and an angle or three sides are given. (B)</p> Signup and view all the answers

What is the formula for the height of a pole using the sine and tangent rules?

<p>h = (d sin(α)) / sin(β) (B)</p> Signup and view all the answers

What is the formula for the area of a triangle with sides a and b, and angle C?

<p>Area = (1/2)ab sin(C) (C)</p> Signup and view all the answers

When should you use the sine rule in solving triangular problems?

<p>When no right angle is given, and either two angles and a side or two sides and an angle are given. (D)</p> Signup and view all the answers

What is the formula for the height of a building using the sine rule?

<p>h = (b sin(α) sin(θ)) / sin(β + θ) (C)</p> Signup and view all the answers

What is the general solution to the equation tan(θ) = x in the interval [0°, 360°]?

<p>θ = tan⁻¹(x) + k × 180° (B)</p> Signup and view all the answers

What is the formula for the area of a triangle with sides a, b, and c?

<p>Area = √(s(s-a)(s-b)(s-c)) (B)</p> Signup and view all the answers

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