write each trigonometric expression as an algebraic expression calculator

Question

cos(arccos(1/2) - arcsin(x))

Answer 1

arccos and arcsin both produce angles so your problem is finding cos(θ₁ - θ₂) which means you would need to use the cosine angle subtraction identity to work this problem.

The cosine angle subtraction identity goes like this:

cos(x – y) = cos(x)cos(y) + sin(x)sin(y)

or, in this case:

cos(θ₁ – θ₂) = cos(θ₁)cos(θ₂) + sin(θ₁)sin(θ₂)

Now, lets find θ₁ and θ₂.

Make a sketch of arccos(1/2) or θ₁:

Because cos(θ) = adjacent/hypotenuse:

and we can calculate "a" by using the Pythagorean Theorem:

1² + a² = 2²

1 + a² = 4

a² = 3

a = √3

That means sin(θ₁) = √3/2

Now make a sketch of arcsin(x) also known as arcsin(x/1) or θ₂:

Because sin(θ) = opposite/hypotenuse:

Again we can calculate "b" by using the Pythagorean Theorem:

b² + x² = 1²

b² = 1 - x²

b = √(1 - x²)

This means cos(θ₂) = √(1 - x²)/1 or √(1 - x²)

So, summarizing:

cos(θ₁) = 1/2

cos(θ₂) = √(1 - x²)

sin(θ₁) = √3/2

sin(θ₂) = x

Putting these into the cosine subtraction identity, we get:

cos(θ₁ – θ₂) = cos(θ₁)cos(θ₂) + sin(θ₁)sin(θ₂)

= (1/2)(√(1 - x²) + (√3/2)(x)

Simplify as desired

Answer 2

cos(arccos(1/2)) - arcsinx

= 1/2 - arcsinx

most would stop here, as there is no way to combine like terms, as they are unlike terms, a pure number and an angle measured in radians or degrees

domain of arcsines is between -90 and +90 degrees, so arcsinx is an angle A such that -90<A<+90 degrees