Full Marks Are Not Necessarily Awarded for a Correct Answer with No Working

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HAEF IB - MATH HL

TEST 4

Trigonometry

Date: 25 January 2018

by Christos Nikolaidis

Paper 1: Without GDC

Name:______

Questions

1. [Maximum mark: 5]

Use the formula for to find the exact value of.

......

2. [Maximum mark: 4]

Given that , find the exact value of .

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3. [Maximum mark: 5]

Solve the equation

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4. [Maximum mark: 7]

(a) Write down the minimum and the maximum value of the expression

[1 mark]

(b) Solve the equation

, [6 marks]

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5. [Maximum mark: 5]

Consider a right-angled triangle ABC with and sides ,, . Given that , and are in arithmetic progression, find the exact values of and .

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6. [Maximum mark: 14]

(a) Show that

[2 marks]

(b) Hence prove, by mathematical induction, that

for all [7 marks]

in the interval [5 marks]

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Paper 2: With GDC

Name:______

Questions

1. [Maximum mark: 5]

In a triangle ABC, =0.5 radians, AB=50cm, AC=40cm. Find the possible values of angle and the smallest possible area of the triangle ABC. [5 marks]

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2. [Maximum mark: 6]

A ship sails from port A on a bearing of 070o for 50 km and reaches point B. From B, it changes its course and sails on a bearing of 010o for another 50 km to point C. Find

(a) The distance of port A to point C. [4 marks]

(b) The bearing of point C from port A. [2 marks]

......

3. [Maximum mark: 6]

The following three-dimensional diagram shows a vertical flagpole AD on horizontal ground. The angles of elevation of the top D from the points B and C are 40o and 35o respectively.

Given that BC = 50m and find the height of the flagpole.

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4. [Maximum mark: 7]

Consider the circle with center O and radius r, as in the follwing diagram. The lines (AB) and (AC) are tagent to the circle at B and C respectively. Let

(a) Show that the area quadrilateral ABOC is [3 marks]

(b) Given that the area of the shaded region is equal to the area of the sector OBDC,

find the value of angle θ in degrees. [4 marks]

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5. [Maximum mark: 10]

Consider two intersecting circles with radii 6 and 11 respectively, as in the diagram below. The distance between their centres A and B is 13. The line segment MN is tangent to both circles.

Find

(a) The length of the line segment MN. [2 marks]

(b) The size of the angle ABN [2 marks]

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6. [Maximum mark: 6]

(a) Show that [4 marks]

(b) Hence find the exact value of [2 marks]

......