Theory question 112
Trigonometry(a) In a diagram, ΔABD is right-angled at B. |AB| = 3 cm, |AD| = 5cm, ∠ACB = 61° and ∠DAC = x°. Calculate, correct to one decimal place, the value of x.
(b) In the diagram, OABCD is a pyramid with a square base of side 2 cm and a slant height of 4cm. Calculate, correct to three significant figures: (i) the vertical height of the pyramid; (ii) the volume of the pyramid.
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(a) tan ∠DAB = BD/AB = 4/3, so ∠DAB = 53.13°. In right-angled triangle ABC, ∠CAB = 90 − 61 = 29°.
x = 53.13 − 29 = 24.1°.
(b) The diagram marks the slant edge as 4 cm. Half the base diagonal = √2 cm.
(i) Height = √(42 − 2) = √14 ≈ 3.74 cm.
(ii) Volume = ⅓ × 22 × 3.742 ≈ 4.99 cm3.
(If 4 cm is taken as the height of a triangular face instead, h = √15 ≈ 3.87 cm and V ≈ 5.16 cm3.)
First find BD = 4 cm with Pythagoras (3-4-5 triangle).