Solve without a calculator. cos(arcsin (3x/2))
I made a triangle but im stuck after that..
arcsin (3x/2) is base angle Ø of a right angled-triangle, so that the opposite side is 3x and the hypotenuse is 2
So find the third side using Pythagoras, that would be
√(4 - 9x^2)
so cos Ø = √(4-9x^2)/2
or
cos (arcsin (3x/2)) = √(4-9x^2)/2
To solve the given expression without a calculator, you can use the concept of triangle trigonometry. Let's go step by step:
1. Start by drawing a right-angled triangle. Let's label one of the acute angles as angle A and the opposite side to angle A as y. The hypotenuse of the triangle will be 2, and the adjacent side will be 3x.
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2 /___|
3x
2. Use the definition of the sine function to find the length of side y:
sin(A) = y / 2
Since we want to find cos(arcsin(3x/2)), we know that the value of arcsin(3x/2) is angle A in the triangle.
Therefore, sin(A) = 3x/2.
Substituting this value in the equation above:
3x/2 = y / 2
Multiply both sides by 2 to isolate y:
3x = y
3. Now, we need to find the length of the adjacent side of angle A, which is 3x. Let's label it as x (to avoid confusion with the unknown variable):
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/ |
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2 /___|
3x
4. Use the Pythagorean theorem to find the length of x:
x^2 + y^2 = 2^2
Substituting the values we know:
x^2 + (3x)^2 = 4
Simplifying:
x^2 + 9x^2 = 4
10x^2 = 4
Divide both sides by 10:
x^2 = 4/10
Simplifying:
x^2 = 2/5
5. Taking the square root of both sides, we get:
x = ± √(2/5)
6. Since we are dealing with a right-angled triangle, where both acute angles have positive cosine values, the answer will be positive. Hence,
x = √(2/5)
7. Finally, we can find cos(arcsin(3x/2)) by using the lengths we obtained:
cos(arcsin(3x/2)) = cos(A)
Since cos(A) = adjacent/hypotenuse, we can substitute the values:
cos(arcsin(3x/2)) = 3x / 2
Therefore, the solution to cos(arcsin(3x/2)) is 3x/2, where x = √(2/5).