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Math 5 SL Advanced Trigonometry Practice Test Questions

1. Consider the following statements

A: log10 (10x) > 0.

B: –0.5 cos(0.5x) 0.5.

C: – 2

π arctan x

2

π.

(a) Determine which statements are true for all real numbers x. Write your answers (yes or no) in the

table below.

Statement (a) Is the statement true for all

real numbers x? (Yes/No)

(b) If not true, example

A

B

C

(b) If a statement is not true for all x, complete the last column by giving an example of one value of x

for which the statement is false.

(Total 6 marks)

2. f(x) = 4 sin2

3x .

For what values of k will the equation f(x) = k have no solutions?

(Total 4 marks)

3. If A is an obtuse angle in a triangle and sin A = 135

, calculate the exact value of sin 2A.

(Total 4 marks)

4. (a) Factorize the expression 3 sin2 x – 11 sin x + 6.

(b) Consider the equation 3sin2x – 11sin x + 6 = 0.

(i) Find the two values of sin x which satisfy this equation,

(ii) Solve the equation, for 0° x 180°.

(Total 6 marks)

5. (a) Write the expression 3 sin2 x + 4 cos x in the form a cos2 x + b cos x + c.

(b) Hence or otherwise, solve the equation

3 sin2 x + 4 cos x – 4 = 0, 0 x 90 .

(Total 4 marks)

6. Solve the equation 2cos2 x = sin 2x for 0 x π, giving your answers in terms of π.

(Total 6 marks)

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7. Consider y = sin9

x .

(a) The graph of y intersects the x-axis at point A. Find the x-coordinate of A, where 0 x π.

(b) Solve the equation sin9

x = –21 , for 0 x 2 .

(Total 6 marks)

8. The diagram shows the graph of the function f given by

f(x) = A sin x2

+ B,

for 0 x 5, where A and B are constants, and x is measured in radians.

0 1 2 3 4 5

2

y

x

(0, 1)

(1,3)

(3, –1)

(5, 3)

The graph includes the points (1, 3) and (5, 3), which are maximum points of the graph.

(a) Write down the values of f(1) and f(5). (2)

(b) Show that the period of f is 4. (2)

The point (3, –1) is a minimum point of the graph.

(c) Show that A = 2, and find the value of B. (5)

(d) Show that f'(x) = cos x2

.

(4)

The line y = k – x is a tangent line to the graph for 0 x 5.

(e) Find

(i) the point where this tangent meets the curve;

(ii) the value of k. (6)

(f) Solve the equation f(x) = 2 for 0 x 5. (5)

(Total 24 marks)