Question 11 of 18beginner🔧 ApplyNumerical3 marks

Evaluate the following: (i) sin⁡60∘cos⁡30∘+sin⁡30∘cos⁡60∘\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ (ii) 2tan⁡245∘+cos⁡230∘−sin⁡260∘2 \tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ (iii) cos⁡45∘sec⁡30∘+cosec⁡30∘\frac{\cos 45^\circ}{\sec 30^\circ + \cosec 30^\circ} (iv) sin⁡30∘+tan⁡45∘−cosec⁡60∘sec⁡30∘+cos⁡60∘+cot⁡45∘\frac{\sin 30^\circ + \tan 45^\circ - \cosec 60^\circ}{\sec 30^\circ + \cos 60^\circ + \cot 45^\circ} (v) 5cos⁡260∘+4sec⁡230∘−tan⁡245∘sin⁡230∘+cos⁡230∘\frac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}

(i)
Answer

1

(ii)
Answer

2

(iii)
Answer

(32\sqrt{2} - 6\sqrt{6})/8

(iv)
Answer

(43 - 243\sqrt{3})/11

(v)
Answer

67/12

Explanation

This question from Exercise 8.2 tests the application of standard trigonometric values for angles 30°, 45°, and 60°. Students must substitute the exact values and simplify systematically. Part (i) simplifies to sin(60°+30°) = sin 90° = 1. Part (v) uses the fundamental identity sin²θ + cos²θ = 1 in the denominator.

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