Compound angles, double angles and R-form flashcards

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Compound angle
An angle written as a sum or difference of two angles, such as A + B or θ − 30°.
Identity (≡)
A statement true for every value of the variable for which both sides are defined, e.g. sin 2A ≡ 2 sin A cos A. An equation is true only for particular values.
R-form (harmonic form) of a cos θ + b sin θ
A single expression R cos(θ ± α) or R sin(θ ± α), with R > 0 and α usually acute.
R in the R-form
The amplitude of the combined wave: R = √(a² + b²).
α in the R-form
The phase shift: the acute angle with tan α = (coefficient paired with sin α) ÷ (coefficient paired with cos α).
sin(A ± B)
sin A cos B ± cos A sin B. (Yes)
cos(A ± B)
cos A cos B ∓ sin A sin B (the sign flips). (Yes)
tan(A ± B)
(tan A ± tan B)/(1 ∓ tan A tan B). (Yes)
sin 2A
2 sin A cos A. (No — derive it)
cos 2A (three forms)
cos²A − sin²A = 2 cos²A − 1 = 1 − 2 sin²A. (No — derive it)
tan 2A
2 tan A/(1 − tan²A). (No — derive it)
cos²A and sin²A in terms of cos 2A
cos²A = ½(1 + cos 2A); sin²A = ½(1 − cos 2A). (No — derive it)
Graph of y = a cos θ + b sin θ
A single sinusoidal wave with the same period as cos θ, amplitude √(a² + b²), shifted horizontally by α.
Right-angled triangle for 5 cos θ − 2 sin θ = R cos(θ + α)
Adjacent 5, opposite 2, hypotenuse √29: R = √29 and tan α = 2/5.
Geometric proof of sin(A + B)
Stack a triangle with hypotenuse 1 and angle B on a line at angle A; the height of the top vertex is sin A cos B + cos A sin B.
Where are the maxima of y = R cos(θ + α)?
Where θ + α = 0°, 360°, …: the curve y = R cos θ shifted α to the left.
Why does a cos θ + b sin θ = c have no solutions when |c| > R?
The horizontal line y = c lies above the maximum R or below the minimum −R of the wave.
How are the double-angle formulae obtained?
Put B = A in the compound angle formulae, e.g. cos(A + A) = cos²A − sin²A; then use sin²A + cos²A = 1 for the other forms of cos 2A.
Which form of cos 2θ do you use in an equation that also contains sin θ?
cos 2θ = 1 − 2 sin²θ, giving a quadratic in sin θ.
Why must you not divide sin 2θ = sin θ by sin θ?
Solutions with sin θ = 0 are lost. Factorise: sin θ(2 cos θ − 1) = 0.
How do you choose the sign of sin(½θ) or cos(½θ)?
From the quadrant of ½θ, not of θ, after using sin²(½θ) = ½(1 − cos θ) or cos²(½θ) = ½(1 + cos θ).
Greatest value of m/(k + R sin(…)), with m > 0 and k > R
m/(k − R): a positive fraction is greatest when its denominator is least.
Exact value of cos 15°?
cos(45° − 30°) = (√2/2)(√3/2) + (√2/2)(½) = (√6 + √2)/4.
Exact value of tan 105°?
tan(60° + 45°) = (√3 + 1)/(1 − √3) = −2 − √3.
sin A = 4/5, A obtuse. Find sin 2A and cos 2A.
cos A = −3/5; sin 2A = 2(4/5)(−3/5) = −24/25; cos 2A = 2(9/25) − 1 = −7/25.
Write sin θ + √3 cos θ as R sin(θ + α).
R cos α = 1, R sin α = √3: R = 2, α = 60°, so 2 sin(θ + 60°).
Range of 8 cos θ + 15 sin θ + 3?
R = √(64 + 225) = 17, so from 3 − 17 = −14 to 3 + 17 = 20.
Solve cos 2θ = cos θ for 0 ≤ θ < 360°.
2cos²θ − cos θ − 1 = 0 ⇒ (2cos θ + 1)(cos θ − 1) = 0 ⇒ θ = 0°, 120°, 240°.
Simplify 2 sin 4x cos 4x and 1 − 2 sin²3x.
sin 8x and cos 6x (double-angle formulae with A = 4x and A = 3x).
Simplify cos⁴x − sin⁴x.
(cos²x + sin²x)(cos²x − sin²x) = cos 2x.
Which of these formulae are in the formulae booklet?
The compound angle formulae are. The double-angle formulae and the R-form are not.
Solving R cos(θ + α) = c for 0 ≤ θ < 360°
Shift the interval to α ≤ θ + α < 360° + α, list every value of θ + α in it, then subtract α.
Rounding α before the last step
Keep α unrounded in the calculator; subtracting a rounded α can change the final answer.
Proving an identity
Work from one side (usually the more complicated) to the other and state the conclusion; never treat it as an equation.
Interval given as 0 ≤ x < 2π
Work in radians, including α (usually to 3 d.p.).
cos(A + B) sign trap
cos(A + B) = cos A cos B − sin A sin B: plus on the left, minus on the right.

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