If 13 sin x + 5= 0 and x є (0º; 270º) calculate the sum of 2x
13sinx = -5
sinx = -5/13
but from you domain, x must be in III
by Pythagoras x^2 + 5^2 = 13^2
x = ± 12 , but in III x = -12
so cosx = -12/13
I don't quite understand what you meant by
"sum of 2x"
x would be appr. 202.62
did you mean "find the sin 2x ??
then
sin 2x = 2sinxcosx = 2(-5/13)(-12/13) = 120/169
To find the value of x in the equation 13sin(x) + 5 = 0, we need to isolate the sin(x) term.
First, subtract 5 from both sides of the equation:
13sin(x) = -5
Next, divide both sides of the equation by 13:
sin(x) = -5/13
Now, we need to find the values of x that satisfy the equation sin(x) = -5/13 in the given interval (0º, 270º).
Since sin(x) is negative, we know that x must lie in the third and fourth quadrants.
To find x, we can use the inverse sine function (also called arcsin or sin^(-1)). However, most calculators only give the principal value of arcsin, which is in the range of -90º to 90º.
To find the other values of x in the desired interval (0º, 270º), we can consider the symmetry property of the sine function.
Since sin(x) = -5/13 in the third quadrant (180º, 270º), we can find its reference angle in the first quadrant. The reference angle (denoted as α) can be found using the equation:
sin(α) = 5/13
Using a calculator or trigonometric tables, we can find the value of α that satisfies the equation.
Once we have the reference angle α, we can determine the values of x in the third and fourth quadrants by using the following formulas:
x3 = 180º + α
x4 = 360º - α
Once we have the values of x3 and x4, we can calculate the sum of 2x by substituting those values and evaluating the expression:
2x = 2(x3 + x4)
I'm sorry, but without the value of α, I'm unable to provide you with the exact solution for the sum of 2x.