Question

How do you evaluate `cos A = 0.7431`?

Answer 1

`angleA=42^@`

Explanation:

`"given "cosA=0.7431`

`"then "angleA=cos^-1(0.7431)=42^@`

Answer 2

`A = pm text{Arc}text{cos} (0.7431) + 360^circ k quad`integer `k`

`A approx pm 42.0038^circ + 360^circ k quad`integer `k`

Explanation:

Cosines, unlike sines, uniquely determine a triangle angle between `0` and `180^circ.`

But we're not given any such constraint here. As far as we know, `A` can be any angle, any real number of degrees or radians.

What's important to remember is the general solution to `cos x = cos a` is `x=pm a + 360^circ k ,` integer `k.`

The principal value of the inverse cosine gives us a particular solution and we apply the recipe to turn it into the general solution.

`cos A = 0.7431 = cos text{Arc}text{cos} (0.7431)`

`A = pm text{Arc}text{cos} (0.7431) + 360^circ k quad`integer `k`

That's the exact, general answer. It gives all the `A`s whose cosine is `0.7431`.

Now we're supposed to get out our calculator and evaluate the inverse cosine. I don't like that part. I got a nice exact answer here and I have to muck it up.

`A approx pm 42.0038^circ + 360^circ k quad`integer `k`