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Olympiad 3

Exercise 7. Prove each

Exercise 7. Prove each of the following identities: (a) 2 + 3 sin 2 x = 5 − 3 cos 2 x (b) (1 + cot 2 x) cos x = csc x cot x (c) sin 4 x − cos 4 x = − cos 2x (d) sin 2 x + tan 2 x sin 2 x = tan 2 x (e) (f) sin x sec x − 1 + sin x sec x + 1 = 2 cot x 3 − 6 cos 2 x sin x − cos x = 3(sin x + cos x) (g) 1 = sec 1 2 x 1 − 1 1 − 1 − sec 2 x (h) 1 − tan2 x 1 + tan 2 x = 2 cos2 x − 1 (i) cos x 1 − sin x − 1 cos x = tan x (j) tan x − 1 tan x + 1 = 1 − cot x 1 + cot x Exercise 8. Given that sin x + sin y = a and cos x + cos y = a where a ≠ 0. Express sin x + cos x in terms of a. Exercise 9. If k = 1 + sin x cos x and sin x in terms of k. , prove that 1 k = 1 − sin x . Hence express cos x cos x √ 1 − sin x 1 + sin x Exercise 10. Prove that = sec x − tan x, where x is acute. Why √ 1 − sin x is the restriction on x necessary? What will the expression be if 1 + sin x (a) x is obtuse? (b) x can take any real value where sin x ≠ −1? Exercise 11. Given √ 2 cos θ = − sin θ and that θ is obtuse, find the value of 1. tan(−θ) 2. sec θ 3. csc(90 ◦ − θ) Exercise 12. Let (a) : y = m 1 x + c 1 and (b) : y = m 2 x + c 2 be the straight lines where m 1 m 2 ≠ 0. Use Trigonometric formulae to prove that (a) and (b) are perpendicular iff m 1 m 2 = −1. Exercise 13. Let A = sin −1 x, where x > 0. Show that cos A = √ 1 − x 2 . Then express in terms of x for csc A and cos 2A. 2 Score:

Exercise 14. Find the exact value of the following: (a) sin 15 ◦ and cos 15 ◦ (c) sin 105 ◦ and cos 105 ◦ (b) sin 75 ◦ and cos 75 ◦ (d) sin 165 ◦ and cos 165 ◦ Exercise 15. Given cos 36 ◦ = 1 + √ 5 , find the exact value of sin 36 ◦ , sin 72 ◦ , 4 sin 18 ◦ , sin 9 ◦ , sin 3 ◦ and sin 6 ◦ . The hint to show: cos 36 ◦ = 1 + √ 5 , to show this, we use the regular 4 pentagon. A B F E C D • show that ∠F EA = 36 ◦ • show that △ABE and △F AE are similarity triangles. • Show that △BAF is an isosceles triangle, hence show that (AB + EF ) × EF = AB 2 • let x = AB EF , then form up the equation x2 − x − 1 = 0, solve for x > 0 • using cos 36 ◦ = AB 2F E = 1 x, replace value of x, that is the exact value. 2 √ 5 Exercise 16. Given sin a + sin b = 3 of cos(a − b). and cos a + cos b = 1, find the value Exercise 17. The area of an equilateral triangle is 100 √ 3 cm 2 . If its perimeter is p cm, find the value of p. 3 Score:

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