Question

Cos(100)-cos(99)=? Can you do it step by step

Answer 1

Let's see what to do buddy...

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If 99 & 100 are in Degree

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STEP (1)

The angles 99 and 100 are in the second quarter of the trigonometric circle.

The cosine is negative in the second quarter.

To make things easier, we can use angle conversion.

Look :

$we \: know \: that \: 100 = 90 + 10$

$also \: we \: know \: 99 = 90 + 9$

So :

$\cos(100) = \cos(90 + 10) = \cos( \frac{\pi}{2} + 10 )$

$\cos(99) = \cos(90 + 9) = \cos( \frac{\pi}{2} + 9 )$

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STEP (2)

Well now we have to do the arc deletion.

To remove the arc, we remove π/2 from the arc.

Remember that every time we remove π/2 from the arc, the trigonometric ratio changes.

That is, if it is a sine, it becomes a cosine, and if it is a cosine, it becomes a sine.

Or if it is a tangent, it becomes a cotangent, and if it is a cotangent, it becomes a tangent.

$in \: second \: quarter \: cosine \: is \: negative \\ \\ \cos( \frac{\pi}{2} + 10 ) = - \sin(10) = - 0.173$

$\cos( \frac{\pi}{2} + 9 ) = - \sin(9) = - 0.156$

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STEP (3)

$\cos(100) - \cos(99) = \\ - \sin(10) - ( - \sin(9) \: ) = \\ - 0.173 - ( - 0.156) = \\ - 0.173 + 0.156 = - 0.017$

And we're done here.

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If 99 & 100 are in Radian

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STEP (1)

First we need to know how many degrees 1 radian is.

The following equation is used to convert degrees to radians or radians to degrees.

$\frac{degree}{180} = \frac{radian}{\pi}$

So we have :

$\frac{d}{180} = \frac{1}{\pi}$

Multiply the sides of the equation by 180 :

$d = \frac{180}{\pi} = \frac{180}{3.14} = 57.32$

So 1 radian is approximately equal to 57 degrees.

And we have :

$100 \: rad \: = 100 \times 57 = 5700 \: deg \\ \\ 99 \: rad \: = 99 \times 57 = 5601 \: deg$

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STEP (2)

Let's move on to deletion.

Look : 5700° = 15 × 360° + 300°

and : 5601° = 15 × 360° + 201°

We know π rad = 3.14 × 57 = 180° deg

So 2π rad = 2 × 180 = 360 ° deg

Then :5700 = 15 × 2 π + 300° = 30 π + 300°

and :5601 = 15 × 2 π + 201° = 30 π + 201°

Remember that deleting 2π is unconditional.

$\cos(30\pi + 300) = \cos(300) = \cos(360 - 60) = \cos(2\pi - 60) = \cos( - 60) = \\ cosine \: eat \: negative \\ \cos( - 60) = \cos(60) = \frac{1}{2}$

$\cos(30\pi + 201) = \cos(201) = \\ \cos(180 + 21) = \cos(\pi + 21) = \\ - \cos(21) = - 0.933$

$\frac{1}{2} - ( - 0.933) = 0.500 + 0.933 = 1.433$

And we're done.

Thanks for watching buddy good luck.

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