Trigonometry Tricks
Trigonometry is a computational aspect of geometry. Using angles and side lengths, trigonometry assists the mathematician in determining values for angles and sides of triangles. Using relationships among the functions sine, cosine and tangent, these values can be determined as well as other values on a circular or triangular field.
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Basic Right Triangle Tricks
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For determining values of a basic right triangle, the rhyme "soak a toe" -- that is, SOH CAH TOA -- helps a student remember the functions for the right triangle. SOH represents the function for sine where sine (S) equals opposite (O) side length divided by the hypotenuse (H) length. CAH represents the cosine function where cosine (C) equals the adjacent (A) side length divided by hypotenuse (H) length. TOA represents the tangent function where tangent (T) equals the opposite (O) side length divided by the adjacent (A) side length.
Wheel Trigonometry Tricks
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All trigonometric functions can be determined by the simple wheel diagram, which contains a circle with one vertical north-to-south line, one diagonally northwest to southeast line and one diagonal northeast to southwest line. Tangent lies at the north portion, then clockwise is secant, cosecant, cotangent, cosine and sine functions. To determine a function, find the corresponding function name, and divide the two functions that lie on the same side of the circle between the function and its inverse. For example, tangent's inverse is cotangent. Secant and cosecant lie between the two on one side and sine and cosine lie between the two on the other side. Sine divided by cosine or secant divided by cosecant both solve the function of tangent.
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Angle Sums and Differences
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Another rhyming song to remember in trigonometry is "sine-cosine, cosine-sine, cosine-cosine, sine-sine." The sum of sine angles (a + b) equals the sine of angle a times the cosine of angle b plus the cosine of angle a multiplied by the sine of angle b. Sine (a + b) = (sine a x cosine b) + (cosine a x sine b). The sum of cosine angles (a + b) equals the cosine of angle a times the cosine of angle b minus the sine of angle a multiplied by the sine of angle b. Cosine (a + b) = (cosine a x cosine b) - (sine a x sine b). For angle differences, simply change the addition and subtraction sign to its opposite.
Sum of the Squares
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The sum of the squares of two different functions can easily be determined by first drawing a circle with two lines running north to south and east to west. Inside the northeast sector, a right triangle should be drawn with the right angle on the east-west line, a 45-degree angle in the middle corner and another 45-degree angle on the northeast side of the circle. Label the hypotenuse 1 and the southern angle is .... To determine any functions for the triangle, the formula is the sine of ... squared plus the cosine of ... squared equals one.
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References
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