Basic Trigonometry: WJEC GCSE Numeracy Higher November 2022 Unit 2 Q5

Numeracy Higher November 2022 Unit 2 Q5

WJECHigherCurrent spec6 marksBasic Trigonometry Circles & Sectors

5. Gareth’s living room is rectangular. It has a doorway close to one of the corners of the room.

Plan view of a rectangular room with the door hinge near one corner and the door’s swing shown dashed

Diagrams not drawn to scale

Gareth’s living room door is 76 cm wide.
The door hinge is 41 cm from the corner of the room.
When the door is fully open, it touches the wall 64 cm from the corner.

Plan view: door hinge 41 cm from the corner, door of width 76 cm swinging to touch the other wall 64 cm from the corner; angle x at the hinge; shaded region made of a sector and a right-angled triangle

Diagram not drawn to scale

The shaded region in the diagram is an unusable part of his living room floor.

By first finding the size of angle \(x\), calculate the area of this unusable part of Gareth’s living room floor. [6]

Mark scheme

Scheme	Marks	Comments
(180 −) tan\(^{-1}\) (64/41) or (180 −) cos\(^{-1}\)(41/76) or (180 −) sin\(^{-1}\)(64/76) or equivalent	M2	Note: angle in triangle = 57.3(…) or 57.4 (º) An equivalent method could include Pythagoras followed by trigonometry Allow an angle of 57(º) from correct working M1 for tan(angle) = 64/41 or cos(angle) = 41/76 or sin(angle) = 64/76 or M1 for unrearranged (or better) correct substitutions into the sine or cosine rules
(\(x\) =) 122.6(…) (º)	A1	Allow an answer of 122.7 (º) or 123(º)
(Unusable area =) \(\dfrac{122.6(\ldots) \times \boldsymbol{\pi} \times 76^2}{360} + \dfrac{41 \times 64}{2}\) (6176.5 to 6201) (1312)	M2	FT ‘their derived 122.6(…)’, but if < 90 then only M2A0 or M1A0 are available M1 for \(\dfrac{122.6(\ldots) \times \boldsymbol{\pi} \times 76^2}{360}\)
= 7488.5 to 7513 (cm\(^2\))	A1	FT for similar range provided their x > 90 and allowing π = 3.14 to 3.142