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 − 2cosθ in the form r sin(θ + φ).
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    r = √(a2+b2) = 3.61; tanφ = b/a, so φ = atan2(b, a) = -2.55 rad.
    r = 3.61, φ = -2.55 rad, so y = 3.61 sin(θ − 2.55)
  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) = 0.381 rad.
    r = 5.39, φ = 0.381 rad, so y = 5.39 sin(θ + 0.381)
  4. Express 4sinθ + 7cosθ in the form r sin(θ + φ).
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    r = √(a2+b2) = 8.06; tanφ = b/a, so φ = atan2(b, a) = 1.05 rad.
    r = 8.06, φ = 1.05 rad, so y = 8.06 sin(θ + 1.05)
2. Find the smallest positive solution of each equation (radians), to 3 significant figures.
  1. Find the smallest positive θ (radians) satisfying 4sinθ + 6cosθ = 5.
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    Write the left side as r sin(θ + φ) with r = 7.21, 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 = 1.39 rad
  2. Find the smallest positive θ (radians) satisfying 3sinθ − 4cosθ = 5.
    Show answer
    Write the left side as r sin(θ + φ) with r = 5.00, φ = -0.927. Then sin(θ + φ) = c/r, so θ = asin(c/r) − φ (or π − asin(c/r) − φ); take the smallest positive value.
    θ = 2.50 rad
  3. Find the smallest positive θ (radians) satisfying 5sinθ + 8cosθ = -6.
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    Write the left side as r sin(θ + φ) with r = 9.43, φ = 1.01. Then sin(θ + φ) = c/r, so θ = asin(c/r) − φ (or π − asin(c/r) − φ); take the smallest positive value.
    θ = 2.82 rad
  4. Find the smallest positive θ (radians) satisfying 6sinθ + 2cosθ = 1.
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    Write the left side as r sin(θ + φ) with r = 6.32, φ = 0.322. Then sin(θ + φ) = c/r, so θ = asin(c/r) − φ (or π − asin(c/r) − φ); take the smallest positive value.
    θ = 2.66 rad
3. Find the maxima and minima (and their positions) to 3 significant figures.
  1. Find the maximum and minimum of y = 2sinθ − 6cosθ − 4, and the θ (radians) at which each occurs.
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    Amplitude r = 6.32, φ = -1.25. 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 = 2.32 at θ = 2.82 rad; ymin = -10.3 at θ = 5.96 rad
  2. Find the maximum and minimum of y = 3sinθ + 5cosθ − 6, and the θ (radians) at which each occurs.
    Show answer
    Amplitude r = 5.83, φ = 1.03. 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 = -0.169 at θ = 0.540 rad; ymin = -11.8 at θ = 3.68 rad
  3. Find the maximum and minimum of y = 4sinθ + 4cosθ + 7, and the θ (radians) at which each occurs.
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    Amplitude r = 5.66, 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 = 12.7 at θ = 0.785 rad; ymin = 1.34 at θ = 3.93 rad
  4. Find the maximum and minimum of y = 6sinθ − 3cosθ − 6, and the θ (radians) at which each occurs.
    Show answer
    Amplitude r = 6.71, φ = -0.464. 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 = 0.708 at θ = 2.03 rad; ymin = -12.7 at θ = 5.18 rad
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