how do i find an inverse trig angle that is not on the unit circle and without using a calculator.

For example:

arcsec(3/5) = ?

Question

how do i find an inverse trig angle that is not on the unit circle and without using a calculator.

For example:

arcsec(3/5) = ?

campbell_st · Answer

well this is a really bad example 

but here is the logic

$$\sec = \frac{1}{\cos}$$

so the question is really saying 

$$\sec(x) = \frac{3}{5} $$

or 

$$\frac{1}{\cos(x) }= \frac{3}{5} $$

take the reciprocal of both sides 

$$\cos(x) = \frac{5}{3}$$

now you could find the angle by unfortunately it doesn't exist... 

since the cos of an angle is always less than 1. 

so before finding x = arcsec(5/3) use some basics. 

sec(x) = 5/3

cos(x) = 3/5

x = arccos(3/5)

this would work.

fellowroot · Answer

Thanks, but then how would you find 

x = arccos(3/5) with no calculator and no unit circle

campbell_st · Answer

well one method would be approximate

$$\cos(45) = 1/\sqrt{2} = 0.707$$

cos(60) = 1/2 

so arccos(0.6) is about half way between the 2 angles 45 and 60... so approximate it at 52.5 degrees. 

or maybe 53

use the exact values to help approximate

campbell_st · Answer

you also know the values for sin, cos and tan of 0 degrees, 30 degrees and 90 degrees... 

the will assist in approximating. 

Thats my best idea... other than looking at a trig table.

fellowroot · Answer

Thank you campbell, but I still need to know without having to approximate.

phi · Answer

In general, you need a calculator.  There are only some small subset of angles that you can figure out using other methods.

mertsj · Answer

first of all, the secant of an angle cannot be 3/5 because the secant is always greater than 1 or less than negative 1 so the answer to that question is does not exist.

mertsj · Answer

Secondly, if you had the secant of an angle that did exist, let us say, perhaps, 5/3, you could draw the triangle and find all 6 functions of that angle but if you wanted the actual angle, you would need either a table of values or a calculator.

fellowroot · Answer

So you are telling me that for some sides you simply cannot find an actual angle without a calculator?

mertsj · Answer

yes.

mertsj · Answer

That's why, in the old days, before calculators, we used to carry books full of tables around with us.