Combining a sine and a cosine into a single wave · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Two waves in, one wave out

Mathematics · Geometry · ages 16-17
Name ______________________   Date ____________

  1. Write 6 cos x as R sin(x + α), with R > 0 and 0° ≤ α < 360°. What is α in degrees?

    Answer: ______________

  2. Matching a single-wave form gives R cos α = 9 and R sin α = 12. Which expression equals R sin(x + α)?

    • 12 sin x + 9 cos x
    • 9 sin x + 12 cos x
    • 9 sin(x + 12°)
    • 12 sin(x + 9°)

  3. Which form can represent every expression a sin x + b cos x for real a and b?

    • R sin(x) + α
    • sin(Rx + α) + R
    • R sin(x + α)

  4. For R > 0 and unrestricted real x, the maximum value of R sin(x + α) is R.

    Circle one:   True   False

  5. Let f(x) = sin x + cos x, with angles in degrees and 0° ≤ x ≤ 30°. At which allowed x is f(x) largest?

    • 30°
    • 45°
    • 0°
    • 90°

  6. For unrestricted real x, what is the minimum value of 2 + 9 sin x + 12 cos x?

    Answer: ______________

  7. Which form equals -5 sin x + 5 cos x for every real x? Angles are in degrees, R > 0 and 0° ≤ α < 360°.

    • 5√2 sin(x + 45°)
    • 5√2 sin(x + 315°)
    • 10 sin(x + 135°)
    • 5√2 sin(x + 135°)

  8. The equation sin(x + α) = 2 has a real solution.

    Circle one:   True   False

  9. Rewrite 3 sin x + 4 cos x = 2.5 as 5 sin(x + α) = 2.5. Which next step correctly isolates the sine factor?

    • sin(x + α) = 2.5, without dividing either side
    • sin(x + α) = 5, after dividing 2.5 by 0.5
    • sin(x + α) = 0.5, after dividing by 5

  10. How many solutions does 10 sin(x + 60°) = 5 have in 0° ≤ x ≤ 180°?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Two waves in, one wave out W1-mt_IwQkNDRWyq-s1

  1. 90 · The amplitude is 6. Matching gives cos α = 0 and sin α = 1, so α = 90°.
  2. 9 sin x + 12 cos x · In the expansion, R cos α multiplies sin x and R sin α multiplies cos x. The coefficients are 9 and 12.
  3. R sin(x + α) · Expanding R sin(x + α) gives a sine term and a cosine term. Match their coefficients to a and b.
  4. True · Sine reaches 1, so the expression reaches R × 1 = R. No sine value exceeds 1.
  5. 30° · Write f(x) = √2 sin(x + 45°). The shifted angle runs from 45° to 75°, where sine increases, so use x = 30°.
  6. -13 · The wave amplitude is √(81 + 144) = 15. The constant 2 shifts the minimum to 2 - 15 = -13.
  7. 5√2 sin(x + 135°) · R = √(25 + 25) = 5√2. Matching requires a negative cosine value and a positive sine value of equal magnitude, so α = 135°.
  8. False · Every real sine value lies between -1 and 1. Since 2 exceeds 1, no real solution exists.
  9. sin(x + α) = 0.5, after dividing by 5 · Divide both sides by the nonzero amplitude 5. The right side becomes 2.5/5 = 0.5.
  10. 1 · The shifted interval is 60° ≤ y ≤ 240°. Only y = 150° satisfies sin y = 0.5 there, giving the single solution x = 90°.
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