Trigonometry Examples

Solve for x 8cos(arcsin(x)) = square root of 64-64x^2
Step 1
Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.
Step 2
To remove the radical on the left side of the equation, square both sides of the equation.
Step 3
Simplify each side of the equation.
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Step 3.1
Use to rewrite as .
Step 3.2
Simplify the left side.
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Step 3.2.1
Simplify .
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Step 3.2.1.1
Multiply the exponents in .
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Step 3.2.1.1.1
Apply the power rule and multiply exponents, .
Step 3.2.1.1.2
Cancel the common factor of .
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Step 3.2.1.1.2.1
Cancel the common factor.
Step 3.2.1.1.2.2
Rewrite the expression.
Step 3.2.1.2
Simplify.
Step 3.3
Simplify the right side.
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Step 3.3.1
Simplify .
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Step 3.3.1.1
Write the expression using exponents.
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Step 3.3.1.1.1
Draw a triangle in the plane with vertices , , and the origin. Then is the angle between the positive x-axis and the ray beginning at the origin and passing through . Therefore, is .
Step 3.3.1.1.2
Rewrite as .
Step 3.3.1.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 3.3.1.3
Simplify by cancelling exponent with radical.
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Step 3.3.1.3.1
Apply the product rule to .
Step 3.3.1.3.2
Raise to the power of .
Step 3.3.1.3.3
Rewrite as .
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Step 3.3.1.3.3.1
Use to rewrite as .
Step 3.3.1.3.3.2
Apply the power rule and multiply exponents, .
Step 3.3.1.3.3.3
Combine and .
Step 3.3.1.3.3.4
Cancel the common factor of .
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Step 3.3.1.3.3.4.1
Cancel the common factor.
Step 3.3.1.3.3.4.2
Rewrite the expression.
Step 3.3.1.3.3.5
Simplify.
Step 3.3.1.4
Expand using the FOIL Method.
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Step 3.3.1.4.1
Apply the distributive property.
Step 3.3.1.4.2
Apply the distributive property.
Step 3.3.1.4.3
Apply the distributive property.
Step 3.3.1.5
Simplify and combine like terms.
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Step 3.3.1.5.1
Simplify each term.
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Step 3.3.1.5.1.1
Multiply by .
Step 3.3.1.5.1.2
Multiply by .
Step 3.3.1.5.1.3
Multiply by .
Step 3.3.1.5.1.4
Rewrite using the commutative property of multiplication.
Step 3.3.1.5.1.5
Multiply by by adding the exponents.
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Step 3.3.1.5.1.5.1
Move .
Step 3.3.1.5.1.5.2
Multiply by .
Step 3.3.1.5.2
Add and .
Step 3.3.1.5.3
Add and .
Step 3.3.1.6
Apply the distributive property.
Step 3.3.1.7
Multiply.
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Step 3.3.1.7.1
Multiply by .
Step 3.3.1.7.2
Multiply by .
Step 4
Solve for .
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Step 4.1
Move all terms containing to the left side of the equation.
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Step 4.1.1
Add to both sides of the equation.
Step 4.1.2
Combine the opposite terms in .
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Step 4.1.2.1
Add and .
Step 4.1.2.2
Add and .
Step 4.2
Since , the equation will always be true for any value of .
All real numbers
All real numbers
Step 5
The result can be shown in multiple forms.
All real numbers
Interval Notation: