Solve the equation
tan^-1(under rootx^2+x)+sin^-1(under rootx^2+x+1)=pi/2
Where tan^-1 inverse of tan
two angles add up to 90 degrees
so they are angles in the same right triangle
the tangent of one of them = opposite /adjacent = a/b
the sin of the other = b/c = b/sqrt(a^2+b^2)
a/b = sqrt (x^2 + x)
b/sqrt(a^2+b^2) = sqrt(x^2 + x + 1)
a^2 = b^2 (x^2+x)= b^2 x^2 + b^2 x
b^2 = (a^2+b^2)(x^2+x+1)
b^2 = b^2 x^2 + b^2 x +b^2)(x^2 + x + 1)
1 = (x^2 + x + 1)(x^2+x+1)
looks like
x^2 + x + 1 = 1
x^2 + x = 0
x(x+1) = 0
x = 0 or x = -1
Thank you
You are welcome. Check arithmetic. I did it fast.
To solve the equation:
1. Let's simplify the equation first. The term "tan^-1" refers to the inverse tangent function, commonly denoted as "arctan" or "atan." Similarly, "sin^-1" refers to the inverse sine function, commonly denoted as "arcsin" or "asin."
2. The given equation is: arctan(sqrt(x^2 + x)) + arcsin(sqrt(x^2 + x + 1)) = π/2
3. We can start solving by applying the property of inverse trigonometric functions, which states that:
arctan(sqrt(x)) + arcsin(sqrt(1 - x)) = π/2
4. Take note that we can use this property because the parts inside the arctan and arcsin functions are equal to each other.
5. By comparing the given equation with the property, we can modify it to match the pattern. Rewrite the equation as:
arctan(sqrt(x^2 + x)) + arcsin(sqrt(1 - (x^2 + x))) = π/2
6. Notice that the term sqrt(x^2 + x) matches the pattern of sqrt(x), and sqrt(1 - (x^2 + x)) matches sqrt(1 - x). By using this pattern, we can rewrite the equation as:
arctan(sqrt(x)) + arcsin(sqrt(1 - x)) = π/2
7. Now we have transformed the given equation into a simpler form - one that matches the property of inverse trigonometric functions.
8. Using the property, we know that:
arctan(sqrt(x)) + arcsin(sqrt(1 - x)) = π/2
This equation holds true if and only if:
sqrt(x) + sqrt(1 - x) = 1
9. Square both sides of the equation:
(sqrt(x) + sqrt(1 - x))^2 = 1^2
Simplify:
x + 2*sqrt(x(1 - x)) + (1 - x) = 1
Further simplification:
x + 1 - x + 2*sqrt(x(1 - x)) = 1
Combine like terms:
2*sqrt(x(1 - x)) = 0
10. Now, to solve for x, we can isolate the square root term by dividing both sides of the equation by 2:
sqrt(x(1 - x)) = 0
11. The square root of a number is equal to zero only when the number itself is equal to zero. Hence:
x(1 - x) = 0
12. This equation can be solved by setting each term to zero:
x = 0 or 1 - x = 0
Solving the second equation (1 - x = 0) gives:
x = 1
13. Therefore, the solution to the equation arctan(sqrt(x^2 + x)) + arcsin(sqrt(x^2 + x + 1)) = π/2 is x = 0 and x = 1.