Trigonometry Answers

cos^(-1)⁡(-x)= - cos^(-1)⁡(x)or cos^(-1)⁡(x) ?? which one is correct?

sec^(-1)⁡(-x)= - sec^(-1)⁡(x)or sec^(-1)⁡(x) ?? which one is correct?

Cos^-1 (-x) = ? please explain the answer?

Sin^(-1)⁡(cos⁡θ )= (π/2) - θ or (π/2) + θ I think both are correct but not sure.?
sin^(-1)⁡(cos⁡θ )= (π/2) - θ or (π/2) + θ
I think both are correct but not sure.

Here my thinking
sin^(-1)⁡(cos⁡θ )=sin^(-1)⁡(sin ((π/2) - θ))=(π/2) - θ


Or in another way
sin^(-1)⁡(cos⁡θ )=sin^(-1)⁡(sin (⁡(π/2)+ θ))= (π/2) + θ)

i want to differentiate sin^(-1)⁡(cos⁡θ ). so what should we differentiate (π/2) - θ or (π/2) +θ.please let me know

2Sin^(-1) x = Sin^(-1)⁡ {2x*√(1-x^2)}⁡ it's okay. but i think it's proof is if we assume x= sin z?
2Sin^(-1) x = Sin^(-1)⁡ {2x*√(1-x^2)}⁡ it's okay. but i think
it's proof is
if we assume x= sin z

sin⁻¹(2sin z*sqrt(1-sin^2 z))
=sin⁻¹(2sin z cos z )
=sin⁻¹(sin 2z)
=2z
=2 sin⁻¹x
2cos^(-1) x = Sin^(-1)⁡ {2x*√(1-x^2)}⁡ it's also okay

it's my thinking
if we assume x= cos z

sin⁻¹(2cos z*sqrt(1-cos^2 z))
=sin⁻¹(2cos z sin z )
=sin⁻¹(sin 2z)
=2z
=2 cos⁻¹x

so now my question is are they both correct?
i want to differentiate this Sin^(-1)⁡ {2x*√(1-x^2)}
so if we use 2 sin^-1 x , then the answer will be 2/√(1-x^2)

and if we use 2 cos^-1 x , then the answer will be -2/√(1-x^2). there are two different answer after differentiate. please let me know.which one is correct or both correct??

If cos(a)=-12/13 and cot(b)=24/7,90<a<180 and 180<b<270 then the quadrant in which a+b lies?

A tree is leaning with a stick with an angle of 30 degree with the tree at a height of 7 m from the foot of the tree. What is the length of the stick? If the distance from the touch point of the stick to the tree is double of the distance from the foot of the tree to the foot of the stick, what is the length of the whole tree?

In triangle ABC prove that a^2 *cot A + b^2 *cot B + c^2 *cot C =4*Delta. where Delta=area of the triangle.

Solve the equation for exact solutions over the interval [0,
2π).



4sin2x+8sinx+4=0

a Boy standing on a horizontal plane finds a bird flying at a distance of 100m from him at an elevation of 30 degree . a girl on the roof of 200 m high building finds the angle of elevation of the same bird to be 45 degree. both the girl and the boy are on the opposite sides of the bird . find the distance of the bird from the girl

Hello there.
And I have a little problem about sine law problems.
And I seem I can't get the right answer.
So here's the question:
As you walk on a straight level path toward a mountain, the measure of the angle of elevation to the peak of the mountain from one point is 33 degrees. From a point 1000 ft closer, the angle of elevation is 35 degrees. How high is the mountain?

Thanks in advance.