solve to four decimal places
1.3224sinx + 0.4732=0 :for all real x values
solve for sinx, then arcsinx.
solve for sinx, then arcsinx.
so its sinx= -.4732/1.3224
and i get -.36594 but how do you get the other values for a real x
Ok, use your calculator. Sinx is negative in the third and fourth quadrants.
from my calc: arcsinx=-20.967
Now the thinking part.
x= 360-20.967 + n360 will give all those angle.
but the other one is
x= 180+20.967 + n360 for the other ones.
thats how you find degrees right? i need to find radian. which is x+2kpi so all i have to do is -.3659+2kpi.
To solve the equation 1.3224sin(x) + 0.4732 = 0 for all real x values:
1. First, isolate the term with the sine function.
Subtract 0.4732 from both sides:
1.3224sin(x) = -0.4732
2. Next, solve for sin(x) by dividing both sides by 1.3224:
sin(x) = -0.4732 / 1.3224
3. Use a calculator or trigonometric table to evaluate sin(x). In this case, sin(x) is negative, so it lies in the third and fourth quadrants.
4. Use the calculator to find the inverse sine (arcsin) of the value calculated in step 3.
arcsin(-0.4732 / 1.3224) ≈ -20.967 degrees
5. Since we want to find the solutions for all real x values, we can use periodicity of the sine function.
In degrees: x = 360 - 20.967 + n * 360 for the first set of solutions, where n is an integer representing the number of full cycles.
In degrees: x = 180 + 20.967 + n * 360 for the second set of solutions.
6. If you need to express the solutions in radians, you can convert. Since sin(x) = -0.4732, the radian solutions can be expressed as:
x = -0.3659 + 2kπ, where k is an integer representing the number of full cycles.
Remember to convert degrees to radians by multiplying by π/180.
So, the general solution to the equation 1.3224sin(x) + 0.4732 = 0, in radians, is x = -0.3659 + 2kπ, where k is an integer.