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.

Two waves in, one wave out

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

  1. Write 4 sin x + 3 cos x as R sin(x + A) with A acute. What is tan A? Give your answer as a decimal.

    Answer: ______________

  2. Write 3 sin x + 4 cos x in the form R sin(x + A) with R > 0. What is R?

    • 7
    • 5
    • 25
    • 12

  3. What form does a sin x plus b cos x take?

    • R plus sin x plus alpha
    • A times b times sin x
    • R sin of x plus alpha

  4. The largest value of R sin of x plus alpha is R.

    Circle one:   True   False

  5. The maximum of 4 sin x + 3 cos x for x between 0 and 360 degrees occurs at x equal to what, to one decimal place?

    • 53.1 degrees
    • 36.9 degrees
    • 143.1 degrees
    • 323.1 degrees

  6. What is the maximum value of 5 sin x + 12 cos x?

    Answer: ______________

  7. What is the maximum value of 6 sin x + 8 cos x?

    • 14
    • 48
    • 100
    • 10

  8. Sin of x plus alpha equals 2 has solutions worth finding.

    Circle one:   True   False

  9. How do you start 3 sin x plus 4 cos x equals 2.5?

    • Square both sides at once
    • Set alpha to zero first
    • Divide by 5 to get sin of x plus alpha equals 0.5

  10. Write 8 sin x + 15 cos x = 4.25 with the left side as R sin(x + A). What is sin(x + A)?

    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. 0.75 · tan A = 3/4 = 0.75. Dividing the cosine coefficient by the sine coefficient gives the tangent.
  2. 5 · R = sqrt(9 + 16) = 5. Squaring and adding the coefficients gives 25, and R is its square root.
  3. R sin of x plus alpha · Two same-frequency waves merge into one shifted wave.
  4. True · Sine peaks at 1, so the product peaks at R.
  5. 53.1 degrees · 53.1 degrees. Here tan A = 0.75 so A is about 36.9 degrees, and the peak needs x + A = 90 degrees.
  6. 13 · The maximum is 13, since sqrt(25 + 144) = 13.
  7. 10 · The maximum is 10. R = sqrt(36 + 64) = 10, and a sine wave peaks at its amplitude.
  8. False · Sine never leaves minus 1 to 1, so there is nothing to solve.
  9. Divide by 5 to get sin of x plus alpha equals 0.5 · Dividing by R leaves a plain sine equation to solve.
  10. 0.25 · sin(x + A) = 0.25. Since R = 17, dividing 4.25 by 17 gives 0.25.
Worksheet · LightMySky