Exercise 2015-12 E J-00010(3 ) (From: NSC Paper 2 Nov 2013)

(1) In the diagram below, reflex T O P=α and P has coordinates (-5 ; -12).

Determine the value of each of the following trigonometric ratios WITHOUT using a calculator:

(a) cos α (3)

(b) tan(180˚ - α) (2)

(c) sin(30˚ - α) (3)

(2) Prove the following identity:

cos2(90°+θ)cos (−θ )+sin (90 °−θ ) cosθ

= 1cosθ - 1 (6)

(3) Determine the general solution of: tan x sin x + cos x tan x = 0 (7)

(4) Consider the following expression: 2 sin23x - sin2x - cos2x

(a) Simplify the expression to a single trigonometric ratio of x. (3) (b) Write down the minimum value of the expression. (1)

Y

X

P(-5 ; -12)

O

α

(5) It is given that p = cos α + sin α and q = cos α - sin α

(a) Determine the following trigonometric ratios in terms of p and/or q:

(i) cos 2α (3) (ii) tan α (4)

(b) Simplify p2q

− q2 p to a single trigonometric ratio of α . (6)

(6) (a) Draw the graphs of f(x) = tan x + 1 and g(x) = cos 2x for x ∈ [-180˚ ; 180˚] on the same system of axes. Clearly show all intercepts with the axes, turning points and asymptotes. (6)

(b) Write down the period of g. (1)

(c) If h(x) = - cos 2(x + 10˚), describe fully, in words, the transformation from g to h. (2)

(d) For which values of x, where x ¿ 0, will f (x) . g(x) ¿ 0? (4)

(7) The Great Pyramid at Giza in Egypt was built around 2 500 BC. The pyramid has a square base (ABCD) with sides 232,6 metres long. The distance from each corner of the base to the apex(E) was originally 221,2 metres.

(a) Calculate the size of the angle at the apex of a face of the pyramid. (For example C E B). (3)

(b) Calculate the angle each face makes with the base. (For example E F G, where EF ⊥ AB in ∆ AEB). (3)

E(apex)

Great Pyramid of Giza in Egypt