Angles, Triangles and Trigonometry - June 2007, Stage 4&5

Problems

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Sine and Cosine

Stage: 4 Challenge Level: Challenge Level:1

The sine of an angle is equal to the cosine of its complement. Can you explain why and does this rule extend beyond angles of 90 degrees?

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Eight Ratios

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Two perpendicular lines lie across each other and the end points are joined to form a quadrilateral. Eight ratios are defined, three are given but five need to be found.

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Two Regular Polygons

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Two polygons fit together so that the exterior angle at each end of their shared side is 81 degrees. If both shapes now have to be regular could the angle still be 81 degrees?

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Wedge on Wedge

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Two right-angled triangles are connected together as part of a structure. An object is dropped from the top of the green triangle where does it pass the base of the blue triangle?

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Over the Pole

Stage: 5 Challenge Level: Challenge Level:1

Two places are diametrically opposite each other on the same line of latitude. Compare the distances between them travelling along the line of latitude and travelling over the nearest pole.

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Flight Path

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Use simple trigonometry to calculate the distance along the flight path from London to Sydney.

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Pythagoras on a Sphere

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Prove Pythagoras' Theorem for right-angled spherical triangles.