Trigonometric Formulae & Identities (Edexcel IGCSE Further Pure Maths): Exam Questions

Figure 2 shows a right pyramid with a horizontal square base of side 8 cm. The vertical height of the pyramid is cm and cm.

Find the exact value of .

Find, to 1 decimal place, the size of the angle between and the plane .

Find, to 1 decimal place, the size of the angle between the plane and the plane .

The midpoint of is and is the point on such that

Show that cm.

Find, to 1 decimal place, the size of angle .

Show that

Hence express in the form where and are integers, giving the value of and the value of ,

(i) Find

(ii) Hence evaluate , giving your answer in the form where , and are integers.

In triangle , = cm, cm and cm

The area of triangle is cm²

Find, to 2 decimal places, the value of

Figure 5 shows a right triangular prism where is a rectangle.

cm

Using a formula from page 2,

show that

Without using a calculator,

show that cm

The angle between the plane and the plane is

Find, in cm to 2 significant figures, the length of

Find, in degrees to one decimal place, the size of the angle between the line and the plane

Using formulae (opens in a new tab), show that

(i)

(3)

(ii)

(1)

Show that

Hence, or otherwise, solve, giving exact values in terms of

for

use algebraic integration to find the exact value of

Triangle is such that

Given that angle is obtuse,

find, in cm to one decimal place, the length of

Using a formula (opens in a new tab), show that

Hence, solve the equation

for

Using a formula given on page 2, solve, giving your solutions as exact values

for

Figure 2 shows triangle

where

(i) Find in terms of and

[2]

(ii) Find, in its simplest form, the exact value of

[1]

Figure 2 shows triangle

where

Find the area of triangle

Figure 2 shows triangle

where

The point lies on and such that , and are collinear.

Use a vector method to find vector as a simplified expression in terms of and

Using a formula (opens in a new tab), show that

Using a formula (opens in a new tab), show that

Hence show that

where , and are integers to be found.

Figure 3 shows part of the curve with equation and part of the curve with equation

Point is the intersection of and as shown in Figure 3

Point is the intersection of with the -axis as shown in Figure 3

Point is the intersection of with the -axis as shown in Figure 3

The finite region , shown shaded in Figure 3, is bounded by the -axis, and

Use calculus to find, in its simplest form, the exact area of

Using the formulae (opens in a new tab) , show that

(i)

(ii)

Show that

Solve, for the equation,

Give your answers to the nearest whole number.

Figure 3 shows a right triangular prism . A cross section of the prism is a triangle in which and radians.

In the prism

and radians.

Show that the volume of the prism is

Figure 3 shows a right triangular prism . A cross section of the prism is a triangle in which and radians.

The volume of the prism is increasing in such a way that the size of and the size of remain constant and the length of , the length of and the length of remain constant.
The lengths of and are each increasing at a constant rate of 0.2cm / s

Find the exact rate of increase, in cm³ / s, of the volume of the prism when the area of the rectangular face is 60 cm²

Using formulae (opens in a new tab) , show that

(i)

(ii)

Show that

Solve, in radians to 3 significant figures, for , the equation

Using calculus, find the exact value of

In triangle , , and

Find, in degrees to one decimal place, the size of

In triangle , , and

The area of triangle is 24 cm²

Find, to 3 significant figures, the value of

Figure 1 shows triangle in which

Find the value of

Give your answer in the form where and are integers to be found.

Using the formulae (opens in a new tab), show that

(i)

[2]

(ii)

[3]

Given that where

use the results from part (a) to show that can be written as

Solve for