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5. Prove the relations— (a.) sin (A +B) . sin (A-B) = ein'A-sin'B ; ~ . sin A + sin 3 A - , (l>.) —,-f q . = tan 2 A ; v ; cosA + cos3A . , 1 -tan 2 (45°-A) . „ . (c.) , — ; —- ;.-■, — t-( = sin 2 A. v ; 1 + tan 2 (45°-A) 6. Find an expression for sin (A +B+ C) in terms of the ratios of A, B, and C. If A+ B + C = 180°, show that— sin 2 A + sin 2 B + sin 2 C =2(1 + cos A cos B cos C). 7. If A be the angle of a triangle, find cos A and cos -| A in terms of the sides. If the sides of a triangle are 12ft., 16ft., and 20ft., find the greatest angle and the area of the triangle. 8. Show how to solve a triangle when two sides and the contained angle are given, and give proofs of the formulae employed. Prove that in any triangle a cos A -\-b cos B = c cos (A —B). 9. Given log 2 = -301, and log 3 = -477, find the logarithms of 6, 15, -18, and T V Find also L sin 60°. 10. A steamer at sea sighted a lighthouse bearing due west, and after the steamer had proceeded on a west-north-west course for sixteen miles the bearing of the lighthouse was found to be south-west: find the distance of the steamer from the lighthouse at the time of each observation, [Given sin 22£° = -3827.] Approximate Cost of Paper.— Preparation, not given; printing (a,173 copies), £27 Us. 6<J.

Authority: John Mackay, Government Printer, Wellington.—lB97. Price 9d. I

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