Please I need urgent help with these questions.

Question

1) What are the solutions of sin (x – π) + 2 = 1 in [0, 2π)?

2) Use your calculator to find arcsin (sin (150 degrees) ). Why didn’t the calculator say 150 degrees?

3) What are the four solutions of x⁴ = 1?

Answer 1

1)

sin (x - π) + 2 = 1

sin (x - π) = -1

sin 3π/2 = -1

x - π = 3π/2

Solve for x.

2)

sin 150 = 0.5

arc sin 0.5 = 30 (the reference angle)

3)

1, -1, i, -i

Answer 2

Hi Wilcox!

The short answer for Question 1 is x = π/2 . For more details, watch this video I made for you!

For the other two questions:

#2: You'll want to consider the range of arcsin (as we did in the video above) to find that there can only be one value for arcsin of 1/2 (namely, 30^°).

#3: Consider the fact that x⁴ = 1 is equivalent to x⁴ –1 = 0. You can then take the following steps to factor the expression:

x⁴ – 1 = 0

(x²)² – (1)² = 0

(x² + 1)(x² – 1) = 0 Difference of squares

( (x)² + (1)² )( (x)² – (1)² ) = 0 Sum of squares and difference of squares

(x + i)(x – i)(x + 1)(x – 1) = 0

Then use the zero product property to solve:

(x + i) = 0 (x – i) = 0 (x + 1) = 0 (x – 1) = 0

x = -i x = i x = -1 x = 1

Therefore, x = {-i, i, -1, 1}.

Feel free to reach out to me if I can be of any further assistance to you!

My best,

Kelsey P.