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| Even and Odd functions |
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| i) A function f(x) is said to be even if f(-x) = f(x) |
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| ii) A function f(x) is said to be odd if f(-x) = -f(x) |
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| e.g., f(x) = x3, f(-x) = (-x)3 = -x3 = -f(x) |
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| f(x) = cos x is even for f(-x) = cos (-x) = cos q = f(x) |
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| f(x) = x cos x is odd for f(-x) = (-x) cos (-x) = -x cos x = -f(x) |
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| f(x) = sin x is odd whereas f(x) = x sin x is even. |
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| Statement: |
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| For all real numbers x and y |
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Let x and y be any two real numbers and let P(x)and Q(y) be the corresponding trigonometric points on the unit circle. In the above
figures we have taken x and y so that  |
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| The coordinates of P(x) are (cosx, sin x) and that of Q(y) are (cos y, sin y) by the definitions of cosine and sine functions. |
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| Now choose a point R on the unit circle so that arc AR has a measure of (x-y) units. Then the trigonometric point with respect to (x-y) is R(x-y) and the corresponding coordinates of R are (cos(x-y),sin(x-y)). The arc length of AR is the same as arc PQ and hence the chord lengths of PQ and AR are same. |
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| i) |AR| = distance from R to A. |
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| Since AR = PQ, we have |
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| Statement: |
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For all values of  |
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| Proof: |
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| For all real values of x. |
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| Proof: |
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| For real values of x |
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| Proof: |
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| Proof: |
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| Proof: |
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| Now put x = y, then |
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| Now put x = y, then |
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=  |
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| Proof: |
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| Adding (1) and (2) |
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| Subtracting (2) from (1) |
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| Adding (3) and (4) |
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| Subtracting (4) from (3) |
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| Proof: |
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| Proof: |
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| For all A and B we have |
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| Let A + B = x, A - B = y, then |
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| Adding (b) and (a), we get |
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| Subtracting (b) from (a), we get |
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| Adding (c) and (d), we get |
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| Subtracting (d) from (c), we get |
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| Proof: |
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