Question

If I was given something like:

sin(arccos(something))

How would i find the exact value of that?
and example we had in class was:
sin(arccos(1/2)) = sin(pi/4) = (sqrt3/2)

but i haven't a clue how we got there.

Answer 1

draw a picture

Answer 2

or use a calculator

Answer 3

y = arccos x

sin² y + cos² y = 1

sin² y + (cos (arccos x))² = 1

sin² y + x² = 1

sin² y = 1 - x²

sin y = sqr (1 - x²)

<=>

sin (arccos x) = sqr (1 - x²)

sin (arccos (- 2/3)) = sqr (1 - (- 2/3)²)

sin (arccos (- 2/3)) = sqr (1 - 4/9)

sin (arccos (- 2/3)) = sqr (9/9 - 4/9)

sin (arccos (- 2/3)) = sqr (5/9)

sin (arccos (- 2/3)) = sqr (5) / 3

Answer 4

starting with
sin(arccos(1/2))

first ignoring the sin

theta= arccos(1/2)

so we draw a picture of this


we can determine side 3 by using pythagorean theorem

once you have all sides, you can solve sin(theta)

Answer 5

Find the missing side using Pythagorean Theorem.

Answer 6

One issue that I'm having is actually arcos (-1/2). You are supposed evaluate the expression without using a calculator.