Products, Sums, Linear Combinations, and Applications

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Sum to Product and Product to Sum Identities

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Key Questions

  • Sum to product formulas are a list of identities wich transforms
    the sum of two trig function to the product of two different trig function.

    #cos(a) + cos(b) = 2cos ((a + b)/2) cos((a-b)/2)#

    #cos(a) - cos(b) = -2sin ((a + b)/2) sin((a-b)/2)#

    #sin(a) + sin(b) = 2sin ((a + b)/2) cos((a-b)/2)#

    #sin(a) - sin(b) = 2cos ((a + b)/2) sin((a-b)/2)#

    You can easily derive them from the addition formulas for trig functions.

  • Main approach to solve a trig equation : Use Trig Transformation Identities to transform it to a product of a few basic trig equations. Solving a trig equation finally results in solving a few basic trig equations.
    Transformation Trig Identities that convert Sums to Products .
    1. cos a + cos b = 2cos (a +b)/2cos (a - b)/2
    2. cos a - cos b = -2sin (a + b)/2sin (a - b)/2
    3. sin a + sin b = 2sin (a + b)/2cos (a - b)/2
    4. sin a - sin b = 2cos (a + b)/2sin (a - b)/2
    5. tan a + tan b = sin (a + b)/cos acos b.
    6. tan a - tan b = sin (a - b)/cos a
    cos b
    Example 1 . Transform f(x) = sin a + cos a to a product.
    Solution. Use Identity (3) to transform f(x) = sin a + sin (Pi/2 - a) = 2sin (Pi/4)sin (a + Pi/4)
    Example 2 . Transform f(x) = sin x + sin 3x + sin 2x to a product. Use Identity (3) to transform the sum (sin x + sin 3x), then put in common factor.
    f(x) = (2sin 2acos a) + 2sin acos a = 2cos a(2sin 3a/2*cos a/2)

  • Here are three very basic examples of using trigonometry

    1. At a point on the ground 50 feet from the foot of a tree, the angle of elevation to the top of the tree is 53º. Find the height of the tree. (Environmental scientists may want to track data on tree height)

    2. From the top of a lighthouse 210 feet high, the angle of depression of a boat is 27º. Find the distance from the boat to the foot of the lighthouse.

    3. An airplane rises vertically 1000 feet over a horizontal distance of 1 mile. What is the angle of elevation of the airplane’s path?

    The last two can be now be calculated with technology but someone had to do the math first in order to invent the technology used.

    As often the case with mathematics the more interesting examples are much more complicated. Engineering, finance, medicine, computer science and programming all use trigonometry. It may be difficult to see real practical use at first but the more you learn the more you realize the applications of trigonometry are everywhere.

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