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 −2sinθ + 4cosθ in the form r sin(θ + φ).
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    r = √(a2+b2) = 4.47; tanφ = b/a, so φ = atan2(b, a) = 2.03 rad.
    r = 4.47, φ = 2.03 rad, so y = 4.47 sin(θ + 2.03)
  3. Express −5sinθ − 2cosθ in the form r sin(θ + φ).
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    r = √(a2+b2) = 5.39; tanφ = b/a, so φ = atan2(b, a) = -2.76 rad.
    r = 5.39, φ = -2.76 rad, so y = 5.39 sin(θ − 2.76)
  4. Express −5sinθ + 7cosθ in the form r sin(θ + φ).
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    r = √(a2+b2) = 8.60; tanφ = b/a, so φ = atan2(b, a) = 2.19 rad.
    r = 8.60, φ = 2.19 rad, so y = 8.60 sin(θ + 2.19)
2. Find the smallest positive solution of each equation (radians), to 3 significant figures.
  1. Find the smallest positive θ (radians) satisfying 4sinθ − 2cosθ = 1.
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    Write the left side as r sin(θ + φ) with r = 4.47, 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 = 0.689 rad
  2. Find the smallest positive θ (radians) satisfying 4sinθ + 5cosθ = -3.
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    Write the left side as r sin(θ + φ) with r = 6.40, φ = 0.896. Then sin(θ + φ) = c/r, so θ = asin(c/r) − φ (or π − asin(c/r) − φ); take the smallest positive value.
    θ = 2.73 rad
  3. Find the smallest positive θ (radians) satisfying 5sinθ − 6cosθ = 3.
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    Write the left side as r sin(θ + φ) with r = 7.81, φ = -0.876. Then sin(θ + φ) = c/r, so θ = asin(c/r) − φ (or π − asin(c/r) − φ); take the smallest positive value.
    θ = 1.27 rad
  4. Find the smallest positive θ (radians) satisfying 8sinθ − 8cosθ = -8.
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    Write the left side as r sin(θ + φ) with r = 11.3, φ = -0.785. 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
3. Find the maxima and minima (and their positions) to 3 significant figures.
  1. Find the maximum and minimum of y = 6sinθ + 7cosθ − 4, and the θ (radians) at which each occurs.
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    Amplitude r = 9.22, φ = 0.862. 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 = 5.22 at θ = 0.709 rad; ymin = -13.2 at θ = 3.85 rad
  2. Find the maximum and minimum of y = 5sinθ − 4cosθ + 6, and the θ (radians) at which each occurs.
    Show answer
    Amplitude r = 6.40, φ = -0.675. 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 = 12.4 at θ = 2.25 rad; ymin = -0.403 at θ = 5.39 rad
  3. Find the maximum and minimum of y = 7sinθ + 3cosθ − 2, and the θ (radians) at which each occurs.
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    Amplitude r = 7.62, φ = 0.405. 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 = 5.62 at θ = 1.17 rad; ymin = -9.62 at θ = 4.31 rad
  4. Find the maximum and minimum of y = 2sinθ + 4cosθ − 8, and the θ (radians) at which each occurs.
    Show answer
    Amplitude r = 4.47, φ = 1.11. 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 = -3.53 at θ = 0.464 rad; ymin = -12.5 at θ = 3.61 rad
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