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IGCSE Math B9.A Sine, cosine and tangentTopic Practice

9.A Sine, cosine and tangent

Use of sine, cosine and tangent of angles up to 180° Angles will be measured in degrees and decimals of a degree

Question 1(a)

[Maximum number: 2]
Question image

Figure 1 shows a quadrilateral ABCD.
AB=9.3 cm,AC=5 cm,AD=10.8 cm.AB=9.3\text{ cm},\quad AC=5\text{ cm},\quad AD=10.8\text{ cm}.∠ACD=124∘,∠ACB=90∘,∠CAB=x∘,∠ADC=y∘.\angle ACD=124^\circ, \quad \angle ACB=90^\circ, \quad \angle CAB=x^\circ, \quad \angle ADC=y^\circ.
Calculate to one decimal place,

the value of x,

Question 3(a)

[Maximum number: 3]
Diagram NOT accurately drawn

Diagram NOT accurately drawn

In Figure 1, A and B are two points on a circle with centre O. The line AC is the tangent to the circle at A and OBC is a straight line.
AC=15 cm,∠OCA=25∘.AC=15\text{ cm},\qquad \angle OCA=25^\circ.

Calculate the area, in cm2\text{cm}^2 to 3 significant figures, of the circle.

Question 5

[Maximum number: 4]
Question image

Diagram NOT accurately drawn.

Figure 1 shows a framework of wooden beams, with ADC a straight line.
AB=8 m,BC=3.5 m,BD=2.5 m,∠ADB=90∘.AB=8\text{ m},\quad BC=3.5\text{ m},\quad BD=2.5\text{ m},\quad \angle ADB=90^\circ.

Question 5(a)(ii)

(a)

Calculate, giving your answer to one decimal place, the size, in degrees, of ∠BAD\angle BAD.

[ 2 ]

Question 5(b)

(b)

A fourth beam DE is added to the framework.
The point E lies on AB and is such that DE is perpendicular to AB.

Calculate the length, in metres to 3 significant figures, of DE.

[ 2 ]

Question 8

[Maximum number: 5]
Figure 2

Figure 2

Diagram NOT accurately drawn

Figure 2 shows a circle ABC with centre O.
AB is a chord of the circle and M is the midpoint of the chord.

Question 8(b)

(a)

AB=10 m and ∠OAB=20∘\angle OAB=20^\circ.
Calculate the length, in m to 3 significant figures, of OA.

[ 2 ]

Question 8(c)

(b)

The chord BC is extended to the point D so that DA is the tangent to the circle at A.
Given that ∠ODA=40∘\angle ODA=40^\circ, calculate, in m to 3 significant figures, the length of AD.

[ 3 ]

Question 20(a)

[Maximum number: 2]
Question image

ABCDEF is a triangular prism.

AC=FD=8 cmBE=AF=CD=12 cm∠ABC=∠FED=53∘∠BAC=∠EFD=90∘\begin{array}{ll} A C=F D=8 \mathrm{~cm} & B E=A F=C D=12 \mathrm{~cm} \\ \angle A B C=\angle F E D=53^{\circ} & \angle B A C=\angle E F D=90^{\circ} \end{array}

Calculate, in cm to 2 decimal places, the length of AB.
cm

Question 9(b)

[Maximum number: 3]
Diagram NOT accurately drawn

Diagram NOT accurately drawn

Figure 3 shows a circle with centre O and radius 10 cm.
The points A and B lie on the circle such that ∠AOB=100∘\angle A O B=100^{\circ} The point C is such that AC and BC are tangents to the circle.

Calculate the length, in cm to 3 significant figures, of BC.

Question 10

[Maximum number: 4]
Figure 2

Figure 2

Figure 2 shows quadrilateral ABCD in which BC=15 cmB C=15 \mathrm{~cm} and AD=40 cmA D=40 \mathrm{~cm}.
The point E on AD is such that BE is perpendicular to AD with AE=20 cmA E=20 \mathrm{~cm} and ∠ABE=25∘\angle A B E=25^{\circ}

Question 10(a)

(a)

Calculate the length, in cm to 3 significant figures, of AB.

[ 2 ]

Question 10(b)

(b)

The point F on BE is such that FC is perpendicular to BE with angle BCF = 20°.

Calculate the length, in cm to 3 significant figures, of FC.

[ 2 ]

Question 28(b)

[Maximum number: 2]
Question image

The diagram shows a ladder of length 5 x metres with one end resting on horizontal ground, OQB, and the other end resting against a vertical wall, OAP.

Initially the two ends of the ladder are at A and at B, where O A=(4 x-3) metres and O B=(3 x+2) metres, and the ladder is shown in the diagram as a solid line.

For safety reasons the ladder is moved so that the two ends of the ladder are now at P and at Q and the ladder is shown in the diagram as a dashed line.
The ladder now makes an angle of 76° with the ground, as shown in the diagram.

Calculate the length, in metres to 3 significant figures, of OP.

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