solve for x in the following equations:
1) sin(arcsin x) = 1
2) 2arcsin x = 1
3) cos(arccos x) = 1/3
think of the definition of inverse trig functions
arcsin x is the angle Ø so that sinØ = x
so let arcsin x = Ø
then we have sin Ø = 1
so if sinØ = 1 and sinØ = x
x = 1
(we could see that right away since something like sin(arcsin 30°) = 30° )
2.
2 arcsin x = 1
arcsin x = 1/2
I know that sin π/6 = 1/2 or sin 30° = 1/2
so x = π/6 ( or 5π/6 , since the sine is also positive in quadrant II )
3. Just like in #1, x = 1/3
Okay thank you for the help on one and three. In my book for number two it says the answer is sin1/2 which is approx. 0.479. I'm not sure how that answer comes about.
My apologies
I solved as if it were 2sin x = 1
Don't know how I missed the "arc" since the whole question deals with it.
Anyway:
2 arcsinx = 1
arcsin x = 1/2
which by definition means
x = sin 1/2
so set your calculator to radians, and take sin .5
you will get .479...
Sure, let's solve each equation step-by-step.
1) sin(arcsin x) = 1
To solve this equation, we need to use the trigonometric identity arcsin(sin α) = α.
So, arcsin x = α.
Now let's rewrite the equation with α:
sin(α) = 1
Since sin(α) = 1 only when α = π/2, we have:
α = π/2
Now substitute back α with arcsin x:
arcsin x = π/2
To solve for x, take the sin of both sides:
x = sin(π/2)
The sine of π/2 is equal to 1, so x = 1.
Therefore, the solution to the equation is x = 1.
2) 2arcsin x = 1
We want to solve for x in this equation.
Divide both sides by 2:
arcsin x = 1/2
Now, using the same trigonometric identity arcsin(sin α) = α, we can rewrite the equation as:
x = sin(1/2)
Since the sine of 1/2 is a non-standard value, we can't simplify it further. The solution to this equation is x = sin(1/2).
3) cos(arccos x) = 1/3
First, using the trigonometric identity arccos(cos α) = α, we can rewrite the equation as:
x = cos(α)
Now, let's rewrite the equation with α:
cos(α) = 1/3
To solve for α, we take the inverse cosine (arccos) of both sides:
α = arccos(1/3)
Now substitute back α with arccos x:
arccos x = arccos(1/3)
To solve for x, we take the cosine of both sides:
x = cos(arccos(1/3))
The cosine of arccos(1/3) simplifies to 1/3, so x = 1/3.
Therefore, the solution to the equation is x = 1/3.