How do you find the exact value of sec(arcsin(45))?

2 Answers
John D.
Jul 22, 2017

45

Explanation:

Let θ=arcsin(45). Then sin(θ)=45.

Therefore:

sin2(θ)=1625

1−sin2(θ)=1−1625

1−sin2(θ)=925

cos2(θ)=925

cos(θ)=±35

Note that θ must be between −π2 and π2 since this is the interval on which arcsinθ is defined. Also note that on the interval −π2 to π2, cosθ is always positive, since that interval covers all of the angles on the right half of the unit circle. Therefore:

cos(θ)=35

sec(θ)=53

sec(arcsin(45))=53

Final Answer

John D.
Jul 22, 2017

53

Explanation:

Alternatively, use the 3-4-5 right triangle as a shortcut to the problem.

![http://thefredeffect.com](useruploads.socratic.org)

We can see that sin(θ)=oppositehypotenuse=45.

Therefore, we can say that θ=arcsin(45).

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

We can also see that cos(θ)=adjacenthypotenuse=35.

Therefore, sec(θ)=1cos(θ)=53

And, since we know θ=arcsin(45), this means that:

sec(arcsin(45))=53

Final Answer

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