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

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 x^4 = 1?

 

Thank you !

Answer 1

(1)

 

sin(x – π) + 2 = 1

sin(x - π) = -1

x - π = sin^-1(-1) = 3π/2 + 2πn (n = 0, ±1, ±2, ±3, ...)

 

x = 3π/2 + π + 2πn = 5π/2 + 2πn

 

To restrict the answer to [0,2π], set n = -1 

 

Then

 

x = 5π/2 - 2π = π/2

 

(2)

 

sin^-1(sin(150°)) =  both 30° and 150°

 

Your calculator probably gave you 30°, but keep in mind that arcsine is multivalued.

 

(3) 

 

x⁴ = 1

x⁴ - 1 = 0

(x² - 1)(x² + 1) = 0

(x - 1)(x + 1)(x² + 1) = 0

 

You have 2 real roots: 1 and -1

You have 2 imaginary roots: i, -i

 

Answer 2

Use the substitution       r = x - π

 

sin(r) + 2 = 1

 

sin(r) = -1

 

r = 3π/2

 

 

Then,

 

x - π = 3π/2

 

You can solve for x from this simple equation.

 

 

2)

 

It calculates sin(150) first.  Sine is positive in the 1st and 2nd quadrants.  150 degrees is in the 2nd quadrant.  However, the angle it uses is the smallest acute angle (reference angle on the unit circle).

 

The reference angle is 30 degrees because 180 - 150 is 30.

 

sin(150) = sin(30) = 1/2

 

arcsin(1/2).  What is angle that makes sine equal to 1/2?  That would be 30 degrees.

 

arcsin(sin(150 degrees)) = 30

 

 

3)

 

Subtract 1 to both sides of the equation.

 

x⁴ - 1 = 0

 

The left side of the equation is a difference in perfect squares.  You can factor it as

 

(x² - 1)(x² + 1) = 0

 

(x - 1)(x + 1)(x² + 1) = 0

 

 

From this factorization, you will have 2 real solutions and 2 complex solutions.  Do you know how to find them from this point?

Answer 3

Hi Elle,

1)

sin(x-pi)+2=1

sin(x-pi)=1-2=-1

sin(x-pi)=-1

in the interval [0, 2pi]

x-pi = 3pi/2 

x= 3pi/2+pi

x= 5pi/2

2) Because sin has an inverse at [-pi/2, pi/2], therefore we need to account for this.

that is we subtract 150 from 180 first before we apply arcsin to sin150

in fact calculator will give 30 degrees

3) x^4=1 ==> x^4-1=0

(x-1)(x+1)(x^2+1)=0

x=1 or x=-1 x^2=-1

x^2=-1 ===> complex solutions

x=i or x=-i

Answer 4

sin (x - π) + 2 = 1

sin (x - π) = -1

x - π = 3π/2

x = 5π/2