Trig - equations - auxilliary angles

Questions on this page address:

2. Solving equations requiring the conversion to auxilliary angle format.

3. Solving problems incorporating the auxilliary angle format.

Simplify the following basic expressions.	1. Write the expression sin x - cos x in the form A sin (x - β). Answer.√2sin (x - π/4).
	2. Write in the form Rsin (θ + α). (α in degrees). Answer.3√2sin (x + 19° 28').
	3. (i) Write f(A) = 4cos A - 3 sin A in the form R cos (A + θ) - express θ to the nearest degree. (ii) Hence sketch f(A) for 0° < A < 360°. Answer.R = 5 and θ = 37°.
	4. Given 2cos(x + β) = cosx - Asinx where 0 ≤ β ≤ π/2, find the values of A and β. Answer.A = √3 and β = 60°.
	5. Sketch the curve by first expressing it in the form A sin(x + α). Answer.A = 2 and α = π/6.
	6. (i) Express sin x + cos x in the form A sin (x + β). (ii) Hence sketch y = sin x + cos x (for -π ≤ x ≤ 2π)? (iii) How many solutions are there for sin x + cos x = 1 within that domain? Answer.(i) A = √2 and α = π/6. (iii) There are 3 solutions in that domain.
	7. What is the maximum value of 5 sin 4t + 12cos 4t? Answer.Max = 13.
	8. (i) Find the coordinates of the points where the maximum and minumum values of occur. (ii) Sketch the curve for 0° ≤ x ≤ 2π . Answer.Max at (11π/6, 4); min at (5π/6, -4).
Solve equations.	9. (i) Write in the form 2cos(x + β) where 0 ≤ β ≤ π/2. (ii) Hence or otherwise solve for 0 ≤ β ≤ 2π. Answer.(i) 2cos(x + π/6) (ii) x = π/6 or x = 3π/2.
	10. (i) Express 3 cosx - 4 sinx in the form A cos (x + α) for some R > 0 and 0 ≤ α ≤ 90° giving α correct to the nearest minute. (ii) Hence solve the equation 3 cos γ - 4 sin γ - 5 = 0 giving your answer to the nearest minute (0° < γ < 360°). Answer.γ = 53° 8'.
	11. Solve (for -2π ≤ x≤ 2π). Answer.x= -7π/4 or π/4.
	12. Solve 6 sinx + 8 cosx + 5 = 0 for 0° < x < 360°. Answer.x = 156°52' or 276° 52'.
	13. Solve for 0° ≤ ψ ≤ 360° (to the nearest minute): 2cos ψ + 4sinψ = 4. Answer.ψ = 90°.
	14. Solve the equation 8sin 2θ + 15 cos 2θ = 17 by first writing it in the form A sin (2θ + β) for 0° < θ < 360°. Answer.θ = 14° 2' or 194° 2'.
	15. Find all angles θ where 0 ≤ θ ≤ 2π for which . Answer.π/3 and π.
Solve problems.	16. A position of a particle moving along a straight line can be described by the equation P = 3 sint + 4cost where t is the time (in seconds) since the particle began to move. (i) Express the equation for P in the form Rcos(t - α) where R > 0 and 0 < α < π/2 giving the value of α in radians to 3 decimal places. (ii) Hence find the values of t, correct to the nearest tenth of a second, to indicate when the particle first reaches its maximum and minimum distances from the origin. Also state those distances. Answer.(i) Max of 5 at 0.644 secs. (ii) Min of -5 at 3.785 secs.
	17. The height of a wave at the beach where a surfing competition is to be held is determined by one of the surfers to be estimated by the equation h = 5cos 0.5t + 12 sin 0.5t where t is time in hours. (i) Another surfer claims it is better to redefine the height equation in the form Asin (0.5t + β). Find the values for A and β which will allow that alternative format for the height equation. (ii) What is the initial height of the tide? (iii) Find the first two values for t (t > 0 hours) when the wave reaches a height of 10 metres. Answer.(ii) Initial height is 12 m. (iii) 10 metres at 2.18 hours and 11.97 hours.
	18.(i) Expressin the form Asin (x - α) where A > 0 and 0 ≤ α ≤ π/2. (ii) Determine the minimum value for . (iii) Solve for 0 ≤ x ≤ 2π. Answer.(ii) Minimum at -2√3. (iii) x = 5π/6 and 11π/6.
	19. A particle moves in a straight line with its position at any time t mins described by x = 3cos 2t + 4sin 2t. (i) Develop an equation in the form A cos (2t + β) to describe its velocity. (ii) Hence find the first two occasions when the particle is at rest. Answer.(i) Vel = 10 cos(2t + 0.643). (ii) At rest at 27.5 secs and at 2 mins 2 secs.
	20. The motion of a particle can be expressed as Asin (t + B). Separating this composite equation into its components results in the form . Find the values of A and B. Answer.A=10 and B = π/4.