Trigonometry  auxilliary angles.
Test Yourself 1.
Questions on this page address: 
1. Simplification of basic expressions. 
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°.


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).
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). 


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 
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. 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.
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. 