Theory question 103
Circle Theorems(a) In the diagram, O is the centre of the circle ABCDE, |BC| = |CD| and ∠BCD = 108°. Find ∠CDE.
(b) Given that tan x = √3, 0° ≤ x ≤ 90°, evaluate ((cos x)2 − sin x)/((sin x)2 + cos x).
Show the model answer
Model answer
(a) Triangle BCD is isosceles: ∠CDB = (180° − 108°)/2 = 36°
BE is a diameter (passes through O), so ∠BDE = 90° (angle in a semicircle)
∠CDE = 36° + 90° = 126°
(b) x = 60°: cos x = ½, sin x = √3/2
Numerator = ¼ − √3/2 = (1 − 2√3)/4; denominator = ¾ + ½ = 5/4
Value = (1 − 2√3)/5 ≈ −0.49
Explanation
In (a), look for the diameter through O that gives a right angle.