AQA GCSE Maths: Pythagoras and Trigonometry

A cuboid box $ABCDEFGH$ABCDEFGH has a rectangular base $ABCD$ABCD with $AB = 6$AB=6 cm and $BC = 8$BC=8 cm. The vertical height of the box is $CG = 5$CG=5 cm. The vertex $G$G is directly above $C$C, and $A$A is the corner of the base diagonally opposite to $C$C.

(a) Show that the length of the diagonal $AC$AC of the base is $10$10 cm. (2 marks)

(b) Work out the length of the space diagonal $AG$AG, giving your answer to $3$3 significant figures. (1 mark)

(c) Work out the angle that the space diagonal $AG$AG makes with the base $ABCD$ABCD, giving your answer to $1$1 decimal place. (2 marks)

In triangle $ABC$ABC, $AB = 8$AB=8 cm, $AC = 11$AC=11 cm and the angle $\angle BAC = 37^{\circ}$∠BAC=37∘.

(a) Work out the length of $BC$BC, giving your answer to $3$3 significant figures. (3 marks)

(b) Work out the area of triangle $ABC$ABC, giving your answer to $3$3 significant figures. (1 mark)

A ship sails from port $P$P on a bearing of $050^{\circ}$050∘ for $12$12 km to reach a point $Q$Q. It then sails due east for $9$9 km to reach a point $R$R.

(a) Work out how far north of $P$P the point $Q$Q is, giving your answer to $3$3 significant figures. (2 marks)

(b) Work out how far east of $P$P the point $R$R is, giving your answer to $3$3 significant figures. (2 marks)

A right-angled triangle has its right angle at $B$B. The hypotenuse $AC = 17$AC=17 cm and one of the shorter sides $AB = 8$AB=8 cm. Work out the length of $BC$BC. (3 marks)

(a) Write down the exact value of $\sin 60^{\circ}$sin60∘. (1 mark)

(b) A right-angled triangle has a hypotenuse of length $12$12 cm and an angle of $30^{\circ}$30∘. Work out the exact length of the side opposite the $30^{\circ}$30∘ angle. (1 mark)

A right-angled triangle has a side of length $9$9 cm opposite an angle $\theta$θ, and the side adjacent to $\theta$θ has length $12$12 cm. Work out the size of angle $\theta$θ, giving your answer to $1$1 decimal place. (1 mark)