Notes & theory for this topic

Trig Functions

Write your working on paper. Reveal each answer only after you've had a go.

1. Transform from a sinθ + b cosθ to r sin(θ + φ). Give angles in radians (−π to π), to 3 significant figures.
  1. Express 3sinθ + 4cosθ in the form r sin(θ + φ).
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    r = √(a2+b2) = 5.00; tanφ = b/a, so φ = atan2(b, a) = 0.927 rad.
    r = 5.00, φ = 0.927 rad, so y = 5.00 sin(θ + 0.927)
  2. Express 3sinθ + 3cosθ in the form r sin(θ + φ).
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    r = √(a2+b2) = 4.24; tanφ = b/a, so φ = atan2(b, a) = 0.785 rad.
    r = 4.24, φ = 0.785 rad, so y = 4.24 sin(θ + 0.785)
  3. Express 7sinθ − 4cosθ in the form r sin(θ + φ).
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    r = √(a2+b2) = 8.06; tanφ = b/a, so φ = atan2(b, a) = -0.519 rad.
    r = 8.06, φ = -0.519 rad, so y = 8.06 sin(θ − 0.519)
  4. Express 5sinθ + 8cosθ in the form r sin(θ + φ).
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    r = √(a2+b2) = 9.43; tanφ = b/a, so φ = atan2(b, a) = 1.01 rad.
    r = 9.43, φ = 1.01 rad, so y = 9.43 sin(θ + 1.01)
2. Find the smallest positive solution of each equation (radians), to 3 significant figures.
  1. Find the smallest positive θ (radians) satisfying 2sinθ + 3cosθ = 3.
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    Write the left side as r sin(θ + φ) with r = 3.61, φ = 0.983. Then sin(θ + φ) = c/r, so θ = asin(c/r) − φ (or π − asin(c/r) − φ); take the smallest positive value.
    θ = 0.000000000000000111 rad
  2. Find the smallest positive θ (radians) satisfying 8sinθ − 7cosθ = 8.
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    Write the left side as r sin(θ + φ) with r = 10.6, φ = -0.719. Then sin(θ + φ) = c/r, so θ = asin(c/r) − φ (or π − asin(c/r) − φ); take the smallest positive value.
    θ = 1.57 rad
  3. Find the smallest positive θ (radians) satisfying 6sinθ − 3cosθ = -1.
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    Write the left side as r sin(θ + φ) with r = 6.71, φ = -0.464. Then sin(θ + φ) = c/r, so θ = asin(c/r) − φ (or π − asin(c/r) − φ); take the smallest positive value.
    θ = 0.314 rad
  4. Find the smallest positive θ (radians) satisfying 2sinθ − 6cosθ = -4.
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    Write the left side as r sin(θ + φ) with r = 6.32, φ = -1.25. Then sin(θ + φ) = c/r, so θ = asin(c/r) − φ (or π − asin(c/r) − φ); take the smallest positive value.
    θ = 0.564 rad
3. Find the maxima and minima (and their positions) to 3 significant figures.
  1. Find the maximum and minimum of y = 7sinθ − 7cosθ + 2, and the θ (radians) at which each occurs.
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    Amplitude r = 9.90, φ = -0.785. The maximum is r + c (when sin = 1, at θ = π/2 − φ); the minimum is −r + c (when sin = −1, at θ = 3π/2 − φ).
    ymax = 11.9 at θ = 2.36 rad; ymin = -7.90 at θ = 5.50 rad
  2. Find the maximum and minimum of y = 6sinθ − 4cosθ + 6, and the θ (radians) at which each occurs.
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    Amplitude r = 7.21, φ = -0.588. The maximum is r + c (when sin = 1, at θ = π/2 − φ); the minimum is −r + c (when sin = −1, at θ = 3π/2 − φ).
    ymax = 13.2 at θ = 2.16 rad; ymin = -1.21 at θ = 5.30 rad
  3. Find the maximum and minimum of y = 8sinθ + 3cosθ − 4, and the θ (radians) at which each occurs.
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    Amplitude r = 8.54, φ = 0.359. The maximum is r + c (when sin = 1, at θ = π/2 − φ); the minimum is −r + c (when sin = −1, at θ = 3π/2 − φ).
    ymax = 4.54 at θ = 1.21 rad; ymin = -12.5 at θ = 4.35 rad
  4. Find the maximum and minimum of y = 5sinθ + 2cosθ − 4, and the θ (radians) at which each occurs.
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    Amplitude r = 5.39, φ = 0.381. The maximum is r + c (when sin = 1, at θ = π/2 − φ); the minimum is −r + c (when sin = −1, at θ = 3π/2 − φ).
    ymax = 1.39 at θ = 1.19 rad; ymin = -9.39 at θ = 4.33 rad
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