Trigonometry for Edexcel IGCSE Further Pure Mathematics: Methods and Worked Examples

Trigonometry is the most extensive section of the Edexcel IGCSE Further Pure Mathematics specification, and it is examined heavily across both written papers. These revision notes cover every required skill, from radian measure through to the addition formulae, with worked examples throughout.

The trigonometry section of the Pearson Edexcel IGCSE Further Pure Mathematics course extends well beyond the basic sine, cosine and tangent work that students encounter at standard IGCSE level. There are seven distinct topic areas within this section, each building on the ones before it. These edexcel igcse further pure mathematics notes provide a thorough treatment of every area, with the kind of worked solutions that make the difference in an exam. If you are looking for edexcel igcse further pure mathematics trigonometry resources, this is the place to start. Whether you are revising for the first time or consolidating what you already know, the igcse 4pm1 trigonometry content below will serve you well. What follows is trigonometry edexcel igcse material explained clearly and formally, with nothing left to guesswork.

Radian measure

The radian is the standard unit of angle measurement in higher mathematics. One radian is the angle subtended at the centre of a circle by an arc whose length equals the radius of that circle. A full circle contains 2pi radians, so pi radians equals 180 degrees.

The conversion formulae are as follows:

• To convert degrees to radians: multiply by pi/180
• To convert radians to degrees: multiply by 180/pi

Two formulae involving radian measure appear regularly in the exam:

Worked example: Arc length and sector area

A sector of a circle has radius 8 cm and angle 1.2 radians. Find the arc length and the area of the sector.

Arc length:

s = r times theta = 8 times 1.2 = 9.6 cm

Sector area:

A = (1/2)r² times theta = (1/2) times 64 times 1.2 = 38.4 cm²

Trigonometric ratios and graphs

The three basic trigonometric ratios (sine, cosine and tangent) must be understood for angles of any magnitude, whether expressed in degrees or radians. The specification requires knowledge of how these functions behave beyond the first quadrant, using the CAST diagram to determine the sign of each ratio in each quadrant.

The CAST diagram divides the coordinate plane into four quadrants, reading anticlockwise from the positive x-axis:

• First quadrant (0 to 90 degrees): All ratios are positive
• Second quadrant (90 to 180 degrees): Only Sine is positive
• Third quadrant (180 to 270 degrees): Only Tangent is positive
• Fourth quadrant (270 to 360 degrees): Only Cosine is positive

The specification also requires that students know the exact values of sine, cosine and tangent for 30, 45 and 60 degrees (and their radian equivalents pi/6, pi/4 and pi/3). These values must be memorised:

Worked example: Exact values

Find the exact value of sin(150 degrees).

150 degrees lies in the second quadrant, where sine is positive.

The reference angle is 180 - 150 = 30 degrees.

Therefore sin(150 degrees) = sin(30 degrees) = 1/2.

Applications in two and three dimensions

Trigonometry questions in two and three dimensions require students to identify right-angled triangles within larger figures and apply the appropriate ratios or rules. In three-dimensional problems, the key skill is finding the angle between a line and a plane, or the angle between two planes.

To find the angle between a line and a plane:

1. Identify the point where the line meets the plane.
2. Drop a perpendicular from another point on the line to the plane.
3. The angle between the line and the plane is the angle in the resulting right-angled triangle at the point where the line meets the plane.

Worked example: Angle in a cuboid

A cuboid has dimensions 6 cm by 4 cm by 3 cm. Find the angle that the space diagonal makes with the base.

Step 1: Find the length of the base diagonal using Pythagoras.

Base diagonal = sqrt(6² + 4²) = sqrt(36 + 16) = sqrt(52) = 2sqrt(13) cm

Step 2: The space diagonal, the base diagonal and the height of the cuboid form a right-angled triangle. The angle theta between the space diagonal and the base satisfies:

tan(theta) = height / base diagonal = 3 / (2sqrt(13))

Step 3: theta = arctan(3 / (2sqrt(13))) = arctan(3 / 7.211) = arctan(0.4160) = 22.6 degrees (to 1 decimal place).

Sine and cosine formulae

The sine rule and cosine rule allow the solution of triangles that are not right-angled. The edexcel igcse further pure mathematics explained approach to these rules requires confident algebraic manipulation.

The sine rule: a/sin(A) = b/sin(B) = c/sin(C)

The cosine rule: a² = b² + c² - 2bc cos(A)

Area of a triangle: Area = (1/2)ab sin(C)

Worked example: The cosine rule

In triangle PQR, PQ = 9 cm, PR = 7 cm and angle QPR = 52 degrees. Find the length QR.

Using the cosine rule with a = QR, b = PR = 7, c = PQ = 9 and A = 52 degrees:

QR² = 7² + 9² - 2(7)(9)cos(52)

QR² = 49 + 81 - 126 times 0.6157

QR² = 130 - 77.58

QR² = 52.42

QR = sqrt(52.42) = 7.24 cm (to 3 significant figures)

Worked example: Area using the sine formula

Find the area of the triangle with sides a = 10 cm, b = 14 cm and included angle C = 35 degrees.

Area = (1/2)ab sin(C) = (1/2)(10)(14) sin(35) = 70 times 0.5736 = 40.2 cm² (to 3 significant figures)

Trigonometric identities

Two fundamental identities are required for the exam:

• cos²(theta) + sin²(theta) = 1
• tan(theta) = sin(theta) / cos(theta)

These identities are used to simplify expressions, prove results and solve equations. The Pythagorean identity can be rearranged to give sin²(theta) = 1 - cos²(theta) or cos²(theta) = 1 - sin²(theta), both of which are frequently needed.

Worked example: Proving an identity

Prove that (1 - cos²(theta)) / sin(theta) = sin(theta).

Left-hand side:

(1 - cos²(theta)) / sin(theta)

= sin²(theta) / sin(theta) (using the Pythagorean identity)

= sin(theta)

= Right-hand side. QED.

Addition formulae

The addition formulae allow the expansion of sin(A + B), cos(A + B) and tan(A + B). These are provided on the formulae sheet in the exam, but fluency with them saves significant time. The six formulae are:

• sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
• sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
• cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
• cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
• tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
• tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))

Worked example: Using the addition formulae

Find the exact value of cos(75 degrees).

Write 75 = 45 + 30.

cos(75) = cos(45 + 30) = cos(45)cos(30) - sin(45)sin(30)

= (1/sqrt(2))(sqrt(3)/2) - (1/sqrt(2))(1/2)

= sqrt(3)/(2sqrt(2)) - 1/(2sqrt(2))

= (sqrt(3) - 1) / (2sqrt(2))

Rationalising the denominator:

= (sqrt(3) - 1) times sqrt(2) / (2 times 2) = (sqrt(6) - sqrt(2)) / 4

Trigonometric equations

Solving trigonometric equations requires students to find all solutions within a given interval. The general approach is:

1. Rearrange the equation so that a single trigonometric function of a single expression is isolated on one side.
2. Find the principal value using inverse trigonometric functions.
3. Use the symmetry of the relevant graph or the CAST diagram to find all solutions in the required range.

Worked example: Solving a trigonometric equation

Solve 2sin²(x) - sin(x) - 1 = 0 for 0 <= x <= 360 degrees.

Step 1: Treat sin(x) as a single variable. Let u = sin(x).

2u² - u - 1 = 0

Step 2: Factorise.

(2u + 1)(u - 1) = 0

u = -1/2 or u = 1

Step 3: Solve sin(x) = -1/2.

The reference angle is 30 degrees. Sine is negative in the third and fourth quadrants.

x = 180 + 30 = 210 degrees, or x = 360 - 30 = 330 degrees.

Step 4: Solve sin(x) = 1.

x = 90 degrees.

Solutions: x = 90, 210 or 330 degrees.

Worked example: Multiple angle equation

Solve cos(2x) = 0.5 for 0 <= x <= 360 degrees.

Step 1: Adjust the range for the substitution. Let y = 2x. Since 0 <= x <= 360, we have 0 <= y <= 720.

Step 2: Solve cos(y) = 0.5 in the range 0 <= y <= 720.

The principal value is y = 60 degrees.

Cosine is positive in the first and fourth quadrants: y = 60, 300, 420, 660 degrees.

Step 3: Divide by 2 to find x.

x = 30, 150, 210, 330 degrees.

Edexcel IGCSE Further Pure Mathematics practice questions for trigonometry

The following self-check questions allow you to test your understanding of the material covered above. Attempt each one without looking back at the edexcel igcse further pure mathematics revision notes, then check your working against the methods shown in the relevant sections.

1. Convert 225 degrees to radians, giving your answer in terms of pi.
2. A sector has radius 5 cm and arc length 12 cm. Find the angle of the sector in radians and hence find its area.
3. Find the exact value of tan(120 degrees) using the CAST diagram and exact values.
4. In triangle ABC, AB = 11 cm, BC = 8 cm and angle ABC = 63 degrees. Find the area of the triangle.
5. Prove that (sin²(theta) + cos²(theta)) / cos(theta) = sec(theta).
6. Use the addition formula to find the exact value of sin(105 degrees).
7. Solve 2cos²(x) + cos(x) - 1 = 0 for 0 <= x <= 360 degrees.

The trigonometry section of the specification is broad and technically demanding, but it is also highly structured. Each topic area connects to the others: radian measure underpins the calculus of trigonometric functions, the identities simplify equations, and the addition formulae extend your ability to find exact values. A student who works through these edexcel igcse further pure mathematics practice questions methodically, correcting errors and identifying patterns, will find that the exam holds few surprises. The Green Bridge CBT platform offers further practice questions organised by topic, allowing you to target specific areas for additional revision.