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Solve for x the equation 3 arcsin(x) = π / 2.
Solution
Divide both sides of the equation by 3.
arcsin(x) = (π / 2) / 3
arcsin(x) = π / 6
Apply sin to both sides and simplify.
sin(arcsin(x)) = sin(π / 6)
The above simplify to
x = 1 / 2
Because of the domain of arcsin(x), we need to verify that the solution obtained is valid.
x = 1 / 2
Right side of equation: 3 arcsin(1 / 2) = 3 (π6) = π / 2.
Left side of equation: π / 2.
The solution to the above equation is x = 1 / 2.
The graphical approximation to the solution to the given equation is shown below. The x coordinate of the point of intersection of the graphs made up the left side and the right of the given equation is 0.5 which is the solution found analytically.
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Solve for x the equation 3 cot (arccos(x)) = 2.
Solution
Divide both sides of the given equation by 3 and simplify.
cot (arccos(x)) = 2 / 3
Let
A = arccos(x)
and apply cos to both sides to obtain.
cos (A) = cos(arccos(x)) = x
Using definition of A above, the equation may be written as.
cot (A) = 2 / 3
Use cot (A) = 2 / 3 to construct a right triangle and find cos(A). Find hypotenuse h first.
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h = √(13)
We now use the same triangle shown above to find cos (A).
x = cos(A) = 2 / √(13) ≈ 0.55
The graphical approximation to the solution to the given equation is shown below.
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Solve for x the equation arcsin(x) = arccos(x).
Solution
Apply sin function to both sides.
sin(arcsin(x)) = sin(arccos(x))
Simplify left side using the identity sin(arcsin(A)) = A.
x = sin(arccos(x))
let A = arccos(x)
cos A = x
sin(arccos(x)) = sin (A) = ~+mn~ √ (1  x^{ 2})
Use the above to rewrite the given equation in algebraic form.
x = ~+mn~ √ (1  x^{ 2})
Square both sides.
x^{ 2} = (1  x^{ 2})
2 x^{ 2} = 1
x = ~+mn~ 1 / √(2)
Because of the domain of the arccos function and we also squared both sides of the equation, we need to verify the solutions and eliminate any invalid (extraneous) ones.
1) x = 1 / √(2)
left side: arcsin( 1 / √(2) ) = π / 4
right side: arccos( 1 / √(2) ) = π / 4
x = 1 / √(2) is a solution to the given equation.
2) x =  1 / √(2)
left side: arcsin(  1 / √(2) ) =  π / 4
right side: arccos(  1 / √(2) ) = 3 &pi / 4
x =  1 / √(2) is not a solution to the given equation.
The graphical approximation to the solution to the given equation is shown below. The x coordinate of the point of intersection 0.71 is close to 1/√2.
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Solve for x the equation arccos(x) = arcsin(x) + π / 2.
Solution
Apply cos function to both sides.
cos(arccos(x)) = cos( arcsin(x) + π / 2 )
Simplify left side using the identity cos(arccos(A)) = A.
x = cos( arcsin(x) + π / 2 )
Expand the right side using the identity cos(a + b) = cos(a).cos(b)  sin(a)sin(b).
x = cos( arcsin(x)) cos(π / 2)  sin( arcsin(x)) sin(π / 2)
Use cos(π / 2) = 0 , sin( arcsin(x)) = x and sin(π / 2) = 1 to simplify the right side of the equation.
x =  x
2 x = 0
x = 0
Verify the solution found.
Left side: arccos(0) = π / 2
Right side: arcsin(0) + π / 2 = π / 2
x = 0 is a solution to the given equation
The graphical approximation to the solution to the given equation is shown below. The x coordinate of the point of intersection is equal to 0 exactly as the value calculated analytically above.
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Solve for x the equation arccos(2x) = π/3 + arccos(x).
Solution
Apply cosine function to both sides.
cos(arccos(2x)) = cos(π/3+ arccos(x))
Simplify left side using the identity cos(arccos(A)) = A and expand the right side using the identity cos(a + b) = cos(a).cos(b)  sin(a)sin(b).
left side: cos(arccos(2x)) = 2 x
right side: cos(π/3+ arccos(x)) = cos(π/3)cos(arcos(x))  sin(π/3)sin(arccos(x))
= cos(π/3)x  sin(π/3)sin(arccos(x))
Rewrite sin(arccos(x)) and the right side of the equation as an algebraic expression.
let A = arccos(x) ,
cos(A) = cos(cos(x)) = x
sin(arcos(x)) = sin(A) = ~+mn~ √ (1  cos^{ 2}A) = ~+mn~ √ (1  x^{ 2})
right side: cos(π/3) x ~+mn~ sin(π/3)√ (1  x^{ 2})
Use cos(π/3) = 1 / 2 and sin(π/3) = √3 / 2 and rewrite the equation using algebraic expressions.
2 x = x / 2 ~+mn~ √3 / 2√ (1  x^{ 2})
Rewrite the equation with radical on the right side.
3 x / 2 = ~+mn~ (√3 / 2) √ (1  x^{ 2})
Square both sides of the equation and simplify.
9 x^{ 2} / 4 = [ ~+mn~ (√3 / 2)√ (1  x^{ 2}) ]^{ 2}
9 x^{ 2} / 4 = (3 / 4)(1  x^{ 2})
Solve for x.
12 x^{ 2} / 4 = 3 / 4
x^{ 2} = 1 / 4
x = ~+mn~ 1 / 2
Because of the domain of the arccos function and we also squared both sides of the equation, we need to verify the solutions and eliminate any invalid (extraneous) ones.
1) x = 1 / 2
left side: cos(arccos(2x)) = cos(arccos(2(1/2))) = cos(arccos(1)) = 0
right side: π/3 + arccos(x) = π/3+ arccos(1 / 2) = π/3 + π/3 = 2 π/3
x = 1 / 2 is not a solution to the given equation.
2) x =  1 / 2
left side: cos(arccos(2x)) = cos(arccos(2(  1/2))) = cos(arccos(  1)) = π
right side: π/3 + arccos(x) = π/3+ arccos( 1 / 2) = π/3 + 2 π/3 = π
x =  1 / 2 is a solution to the given equation.
The graphical approximation to the solution to the given equation is shown below. The point of intersection has an x coordinate equal to 0.5 which is exactly the solution found analytically.
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