trigonometry

Question

find all solutions in radians to 5sin(x-2)=6/10 for 0<x<2 pi

Answer 1

First let's solve the equation for sin(x-2). Simplify the right-hand side to 3/5, then divide both sides by 5:

 

sin(x-2) = 3/5 ÷ 5/1 = 3/5 × 1/5 = 3/25

 

Now, let's take the inverse sine of both sides:

 

x-2 = sin^-1(3/25) = 0.1203 radians (rounded to the nearest ten-thousandth)

 

Since you want to include any other angles where sine is 3/25, you need to think about other quadrants where sine is positive. Sine, which you can think of as a point's y-coordinate, is positive in Quadrant I (between 0 and pi/2 radians) and in Quadrant II (between pi/2 and pi radians).

 

0.1203 radians is in Quadrant I, so you'll need to find its equivalent in Quadrant II by reflecting it over the y-axis. This will give you pi - 0.1203 = 3.0213 radians (rounded to the nearest ten-thousandth).

 

So, x-2 = 0.1203 radians or 3.0213 radians. Solve for x by adding 2 to both sides:

 

x = 2.1203 radians or 5.0213 radians

 

Both of these values are between 0 and 2 pi.

Answer 2

5sin(x - 2) = 6 / 10

 

5sin(x - 2) = 3 / 5

 

 

Divide both sides of equation by 5.

 

sin(x - 2) = 3 / 25

 

 

We substitute  q = x - 2 to make the trig equation easier to deal with.

 

sin(q) = 3 / 25

q = sin^-1 (3 / 25)

 

q = 6.892              and          q = 180 - 6.892

                                            q = 173.108

 

 

Put these values of q into the substitution.

 

q + 2 = x

 

6.892 + 2 = x           and           173.108 + 2 = x

 

      8.892 = x                             175.108 = x

 

 

Multiply these values of x by (π / 180) to get pi radians.

 

8.892(π / 180) = 0.049π

 

175.108(π / 180) = 0.973π

 

x = 0.049π

x = 0.973π