Theory question 108
Trigonometry(a) Given that sin x = 0.6 and 0° ≤ x ≤ 90°, evaluate 2 cos x + 3 sin x, leaving your answer in the form m/n where m and n are integers.
(b) In the diagram, a semi-circle WXYZ with centre O is inscribed in an isosceles triangle ABC. If |AC| = |BC|, |OC| = 30 cm and ∠ACB = 130°, calculate, correct to one decimal place, (i) radius of the circle; (ii) area of the shaded portion. (Take π = 22/7)
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(a) cos x = √(1 − 0.36) = 0.8. 2(0.8) + 3(0.6) = 3.4 = 17/5.
(b)(i) OC bisects ∠ACB, so ∠OCY = 65°. The radius OY is perpendicular to the tangent CB: r = 30 sin 65° ≈ 27.2 cm.
(ii) OA = 30 tan 65° ≈ 64.3 cm. Area of ΔABC = ½ × 2(64.34) × 30 ≈ 1930.1 cm2.
Semicircle = ½ × 22/7 × 27.192 ≈ 1161.7 cm2.
Shaded area ≈ 1930.1 − 1161.7 ≈ 768.4 cm2 (767.5 cm2 if r = 27.2 is used).
The radius to a point of contact is perpendicular to the tangent.