Question

Find the value of sin 23° using appropriate methods.
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Answer 1

$\large\underline{\sf{Solution-}}$

$\rm :\longmapsto\:sin23 \degree$

in radians can be represented as

$\rm \: = \: sin\dfrac{23\pi}{180}$

We use concept of differentiation to find the approximate value of ain23°.

Let we assume that

$\rm :\longmapsto\:f(x) = sinx \: \: where \: x \: = \dfrac{\pi}{6}$

So,

$\rm :\longmapsto\:f(x + h) = sin(x + h) \: \: where \: h \: = - \dfrac{7\pi}{180}$

Using Approximations, we have

$\rm :\longmapsto\:f(x + h) = hf'(x) + f(x)$

On substituting the values, we get

$\rm :\longmapsto\:sin(x + h) = h \: cosx + sinx$

$\red{\bigg \{ \because \:as \: f(x) = sinx \: \: so \: \: f'(x) = cosx\bigg \}}$

Now, on substituting the values of x and h, we have

$\rm :\longmapsto\:sin\bigg(\dfrac{23\pi}{180} \bigg) = - \dfrac{7\pi}{180}cos\dfrac{\pi}{6} + sin\dfrac{\pi}{6}$

$\rm :\longmapsto\:sin\bigg(\dfrac{23\pi}{180} \bigg) = - \dfrac{7\pi}{180} \times \dfrac{ \sqrt{3} }{2} + \dfrac{1}{2}$

$\rm :\longmapsto\:sin\bigg(\dfrac{23\pi}{180} \bigg) = -7 \times \dfrac{3.14}{180} \times \dfrac{1.732}{2} + 0.5$

$\rm :\longmapsto\:sin\bigg(\dfrac{23\pi}{180} \bigg) = - 0.1058 + 0.5$

$\rm :\longmapsto\:sin\bigg(\dfrac{23\pi}{180} \bigg) = 0.3942$

$\bf\implies \:sin23 \degree \: = \: 03941$

Additional Information :-

$\red{\boxed{ \rm{ \frac{d}{dx}sinx = cosx}}}$

$\red{\boxed{ \rm{ \frac{d}{dx}cosx = - \: sinx}}}$

$\red{\boxed{ \rm{ \frac{d}{dx}cosecx = - \: cosecx \: cotx}}}$

$\red{\boxed{ \rm{ \frac{d}{dx}secx = \: secx \: tanx}}}$

$\red{\boxed{ \rm{ \frac{d}{dx}tanx = \: sec^{2} x \:}}}$

$\red{\boxed{ \rm{ \frac{d}{dx}cotx = \: - \: cosec^{2} x \:}}}$