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  • Curriculum and the Dream Paradigm

    - Stephen Schafer

     In addition to the fact that astrology and trigonometry share the fundamental principles of geometry, of course, the concept of proportion is central to trigonometric formulation: 

    "Two triangles are said to be similar if one can be obtained by uniformly expanding the other. This is the case if and only if their corresponding angles are equal, and it occurs for example when two triangles share an angle and the sides opposite to that angle are parallel. The crucial fact about similar triangles is that the lengths of their sides are proportionate. That is, if the longest side of a triangle is twice that of the longest side of a similar triangle, say, then the shortest side will also be twice that of the shortest side of the other triangle, and the median side will be twice that of the other triangle. Also, the ratio of the longest side to the shortest in the first triangle will be the same as the ratio of the longest side to the shortest in the other triangle.

    Figure 6: trigonometric triangle

    Using these facts, one defines trigonometric functions, starting with right triangles, triangles with one right angle (90 degrees or π/2 radians). The longest side in any triangle is the side opposite the largest angle.

    Because the sum of the angles in a triangle is 180 degrees or π radians, the largest angle in such a triangle is the right angle.

    The longest side in such a triangle is therefore the side opposite the right angle and is called the hypotenuse. Pick two right angled triangles which share a second angle A. These triangles are necessarily similar, and the ratio of the side opposite to A to the hypotenuse will therefore be the same for the two triangles. It will be a number between 0 and 1, because the hypotenuse is always larger than either of the other two sides, which depends only on A; we call it the sine of A and write it as sin (A), or simply sin A. Similarly, one can define the cosine of A as the ratio of the side adjacent to A to the hypotenuse.

    These are by far the most important trigonometric functions; other functions can be defined by taking ratios of other sides of the right triangles but they can all be expressed in terms of sine and cosine. These are the tangent, secant, cotangent, and cosecant."

    Figure 8


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