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| Co-terminal Angles |
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| The angles (2p + A), (4p + A)......have the same initial and terminal arm as the angle A and so these angles are called co-terminal angles. For all such angles, the value of any trigonometric ratio is the same. Thus, all co-terminal angles are trigonometrically equivalent. |
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| Let us consider the simplest form of a trigonometric equation, where a trigonometric function is equal to a constant. For example |
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Since the trigonometric functions of all
co-terminal angles are same, and since an angle may have infinite number of
co-terminals, therefore the equation can have infinite number of
roots. |
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| For instance, |
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| All these solutions put together in a compact form or forms are called complete solution or general solution of the equation. |
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| To find the general solution of the following equations |
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| (i) sin q = 0 (ii) tan q = 0 (iii) cos q = 0 (iv) cot q = 0. |
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| Suggested answer: |
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| General solution of (i) sin q = 0 (ii) tan q = 0 |
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| From the figure, we have |
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| This is possible only if the terminal side of the angle q falls in line with OX or OX'. |
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i.e., when q = 0, p, 2p, ........np, where n  Z. |
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| i.e., 0 = np |
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| \ q = np is a general solution of the equation sin q = 0. |
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| values of q, for which y = 0. Hence the general solution of tan q = 0 is |
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| q = np. |
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| iii) General solution of cos q = 0 |
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 cos q = 0 only when x= 0 i.e., the terminal side of q falls in line with OY or OY ' (or coincides with OY or OY ') i.e., when |
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| iv) cot q = 0 |
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| From the figure, we have |
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| cot q = 0 only when x = 0. |
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| When the terminal side of q falls in line with OX or OX' (coincides with OX or OX'), i.e., when |
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| Note: |
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| The equation sec q = 0, cosec q = 0 can have no solutions as the values of sec q and cosec q cannot be numerically less than 1. |
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