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How do you find the exact value of #sin(tan^-1 -5/7)#?

Answer 1

Caroline Adams

#-5/sqrt74#

#tan^(-1)(-5/7)= sin^(-1)(-5/sqrt(5^2+7^2))=sin^(-1)(-5/sqrt74)#

The given expression is

#sin(sin^(-1)(-5/sqrt74))=-5/sqrt74#.

Note that , if arc tan is negative, the angle is in #Q_4#, and so,

#arc sin in Q_4# is also negative.

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Answer 2

Oliver Atkinson

To find the exact value of sin(tan^-1(-5/7)), we can use the relationship between sine and tangent in a right triangle. Let's denote the angle whose tangent is -5/7 as θ. This means that tan(θ) = -5/7.

Now, construct a right triangle where one angle is θ and the opposite side is -5 and the adjacent side is 7 (since tangent is opposite over adjacent in a right triangle).

Then, use the Pythagorean theorem to find the length of the hypotenuse.

Once you have the lengths of all three sides of the triangle, you can find the sine of the angle θ using the definition of sine in a right triangle, which is opposite over hypotenuse.

Finally, you can calculate the exact value of sin(tan^-1(-5/7)) using the values you found in the triangle.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.