How do you find the exact value of sec(arcsin(4/5))?

2 Answers
Jul 22, 2017

4/5

Explanation:

Let theta = arcsin(4/5). Then sin(theta) = 4/5.

Therefore:

sin^2(theta) = 16/25

1 - sin^2(theta) = 1 - 16/25

1 - sin^2(theta) = 9/25

cos^2(theta) = 9/25

cos(theta) = +-3/5

Note that theta must be between -pi/2 and pi/2 since this is the interval on which arcsintheta is defined. Also note that on the interval -pi/2 to pi/2, costheta is always positive, since that interval covers all of the angles on the right half of the unit circle. Therefore:

cos(theta) = 3/5

sec(theta) = 5/3

sec(arcsin(4/5)) = 5/3

Final Answer

Jul 22, 2017

5/3

Explanation:

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

http://thefredeffect.com

We can see that sin(theta) = "opposite"/"hypotenuse" = 4/5.

Therefore, we can say that theta = arcsin(4/5).

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

We can also see that cos(theta) = "adjacent"/"hypotenuse" = 3/5.

Therefore, sec(theta) = 1/cos(theta) = 5/3

And, since we know theta = arcsin(4/5), this means that:

sec(arcsin(4/5)) = 5/3

Final Answer