Small Angle Approximations · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Tiny angles made simple

Mathematics · Geometry · ages 17-18
Name ______________________   Date ____________

  1. Using cos x approx 1 minus x squared over 2, estimate cos(0.2).

    Answer: ______________

  2. Fred says small angle approximations work in degrees too, so sin(1 degree) is about 1. Is Fred right?

    Circle one:   True   False

  3. Using cos x approx 1 - x squared/2, estimate cos(0.2).

    Answer: ______________

  4. For small x measured in radians, sin x is approximately what?

    • 1 minus x squared over 2
    • x
    • 1

  5. Near the origin, how do the graphs of y = sin x and y = x compare?

    • they cross at right angles
    • they are almost identical
    • they stay one unit apart
    • they meet only at x = 1

  6. Near the origin, how do the graphs of y equals sin x and y equals x compare?

    • They cross at right angles
    • They stay one unit apart
    • They are almost identical

  7. Using cos x approx 1 - x squared/2, estimate cos(0.1).

    Answer: ______________

  8. Using cos x approx 1 minus x squared over 2, estimate cos(0.1).

    Answer: ______________

  9. Using sin x approx x, what is (sin 0.06) over 0.06 to the nearest whole number?

    Answer: ______________

  10. Take sin(0.3) approx 0.3. The next Taylor term is -x cubed/6. About how big is the correction?

    • 0.09
    • 0.027
    • 0.00045
    • 0.0045
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Answer key

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

Tiny angles made simple W1-mt_fIsZes5Os3-s1

  1. 0.98 · 0.2 squared is 0.04, half is 0.02, and 1 minus 0.02 is 0.98.
  2. False · Fred is wrong on both counts. The approximations need radians, and sin(1 degree) is about 0.0175.
  3. 0.98 · 0.98. Since 0.2 squared is 0.04, half is 0.02, and 1 - 0.02 = 0.98.
  4. x · The sine graph hugs the diagonal y equals x near the origin.
  5. they are almost identical · They are almost identical. Tangency at the origin keeps them glued together nearby.
  6. They are almost identical · Tangency at the origin keeps them glued together nearby.
  7. 0.995 · 0.995. Since 0.1 squared is 0.01, half is 0.005, and 1 - 0.005 = 0.995.
  8. 0.995 · 0.1 squared is 0.01, half is 0.005, and 1 minus 0.005 is 0.995.
  9. 1 · The approximation makes the fraction about 0.06 over 0.06, which is 1.
  10. 0.0045 · 0.0045. Note 0.3 cubed is 0.027, and dividing by 6 gives 0.0045.
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