I. THE PYTHAGORAS' THEORM
1. Simplify the following:
(a) sin7x.cos6x - cos7x.sin6x (b) 2sin3q.cos3q
2. Simplify the following:
3. Evaluate the following:
4. Find the exact value of the following:
5. If A and B are acute and sinA = 4/5 and cosB=12/13, determine the values of :
(a) cosA (b) sin2A (c) sin(A-B)
6. If x is acute and cosx = 1/3, find the exact values of:
(a) cos(2x) (b) cos(3x) (c) tan(x/2)
7. If tan(x) = 1/2 and x+y = p/4, find the values of :
(a) tan(2x) (b) tan(3x) (c) tan(y)
8. Write down the result of cos(x + y) hence
(a) simplify (b) show that
(c) prove that
9. (i) Assuming results for cos(a + b), cosq, sin2q , find an expression for cos3b in terms of cosb.
If for all values of q, find the values of the constants a, b.
(ii) Write down the result for tan2q in terms of tanq . If tan2q = 5/12, where 0<2q<p/2, find, the value of tanq.
10. (i) Write down the result for tan(a - b) and hence show that tan15 = 2
(ii) Show that = tanx , and use this result to obtain tan(p/12) in simplest form.
11. (i) Prove that = tanx and hence find the value of tan(p/8) in simplest surd form.
(ii) Show that , where , and prove that cos67.5=.
12. Show that cos(x + y) + cos(x - y) = 2cosx.cosy and hence find an expression for
(a) cos75.cos35 in thr form
(b) cos75 + cos35 in the form 2cosA.cosB
13. Find an expression for
(i) sin25.cos10 in the form
(ii) cos60 - cos80 in the form 2cosA.cosB
14. (i) Write down an expression for sin(a + b) and hence show that
If , evaluate
(ii) Prove that where , and hence show that .
15. (i) Starting with the result for cos(x + y), show that 1 - cos2x = 2.sin2x
(ii) Determine the following limits as
(a) (b) (c) (d) .
16. Use the result of to prove that
17. Prove the following identities:
18. Show that
19. Prove each of the following:
20. Show that
21. (i) Show that is independent of . What is its value ?
(ii) If is acute and , where x is so small that is negligible compared with unity, prove that
22. (i) If , , show that
(ii) If and , find an equation linking x, y which is independent of .
(iii) Eliminate in each of the following: