MSI Trigonometry Questions - GIFS

[Pages:28]MATHEMATICS

MATERIAL FOR GRADE 12

Trigonometry

Q U E S T I O N S

QUESTION 1

In the diagram below, ABC is an isosceles triangle. D lies on BC. AB AD .

A

a

a

b

B

D

b

1.1 Determine, without reasons, the size of AC in terms of .

1.2 Prove that: 2

cos 2 22 1

1.3 Hence, determine the value of if 3 and 2 (Rounded off to two decimal digits.)

C

(2)

(4)

(3) [9]

QUESTION 2

Simplify the following without using a calculator.

2.1 cos 56? cos 26? cos 146? sin 26?

(4)

tan(180q x) cos(360q x)

(6)

2.2

sin(x 180q) cos(90q x) cos(720q x) cos(x)

cos 2x cos2 x 3sin 2 x 1

Prove the identity : 2.3

2 2 sin 2 x

cos2 x

(5)

[15]

QUESTION 3 Consider the function f(x) = sin2x for x [90q; 90q]

y

2

1

-90? -75? - 60? -45? -30? -15?

-1 -2

f

15? 30? 45? 60? 75? 90? x

3.1 Write down the period of f.

(1)

3.2 Sketch the graph of g(x) cos(x 15q) for x [ 90q ; 90q] on the

diagram sheet provided for this sub-question.

(5)

3.3 Solve the equation: sin 2x cos(x 15q) for x [ 90q ; 90q]

(7)

3.4 Find the values of x for which f(x) < g(x).

(3)

[16]

QUESTION 4

4.1.1 Simplify the following expression to a single trigonometric function:

2 1 0?

0?

(5)

4.1.2 For which value(s) of x, x 0?; 360? is the expression in 4.1 undefined?

(3)

4.2 Evaluate, without using a calculator:

347?. 1 3? 315? . 64?

4.3 Prove the following identity: 3 2cos2x 1

(5) (5)

[18]

f(x)=-2cos(x) f(x)=sin(x+30)

QUESTION 5 The graphs of f(x) = 2cosx and g(x) = sin(x +30?) for x [ 90?; 180? are drawn in the diagram below.

y

2

1

-90

-60

P

-30 -1

Q

x

30

60

90

120

150

180

g(x) = sin(x+30?)

-2 f(x) = -2cosx

5.1 Determine the period of g.

(1)

5.2 Calculate the x-coordinates of P and Q, the points where f and g intersect.

(7)

5.3 Determine the x-values, x [ 90?; 180? , for which:

5.3.1 g(x) f(x)

(3)

5.3.2 f (x).g(x) 0

[14]

QUESTION 6

AB is a vertical tower of p units high. D and C are in the same horizontal plane as B, the foot of the tower. The angle of elevation of A from D is x. B = y and DB = . The distance between D and C is k units.

6.1.1 Express p in terms of DB and x.

(2)

6.1.2 Hence prove that: p =

(5)

6.2 Find BC to the nearest meter if x = 51,7?, y = 62,5?, 80 m and k = 95 m. (4) [11]

QUESTION 7 In the diagram below, P ( 15; m) i a i i he hi d ad a a d 17c

+ 15 = 0.

y

O

x

.P ( 15 ; m)

7.1 WITHOUT USING A CALCULATOR, determine the value of the following:

7.1.1 m

(3)

7.1.2 sin + a

(3)

7.1.3 c 2

(3)

7.2 Simplify:

sin(180q x).cos(x 180q ).tan(360q x)

sin(x).cos(450q x)

(7)

7.3 Consider the identity:

sin x sin 2x 1 cos x cos2x

tan x

7.3.1 Prove the identity.

(5)

7.3.2 Determine the values of x for which this identity is undefined.

(4)

[25]

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