Trigonometry Examples

8 cos (arcsin (x)) = \sqrt{64 - 64 x^{2}}

$8 \cos (\arcsin (x)) = \sqrt{64 - 64 x^{2}}$

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.

\sqrt{64 - 64 x^{2}} = 8 cos (arcsin (x))

$\sqrt{64 - 64 x^{2}} = 8 \cos (\arcsin (x))$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

{\sqrt{64 - 64 x^{2}}}^{2} = {(8 cos (arcsin (x)))}^{2}

${\sqrt{64 - 64 x^{2}}}^{2} = {(8 \cos (\arcsin (x)))}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{64 - 64 x^{2}}

$\sqrt{64 - 64 x^{2}}$ as

{(64 - 64 x^{2})}^{\frac{1}{2}}

${(64 - 64 x^{2})}^{\frac{1}{2}}$ .

{({(64 - 64 x^{2})}^{\frac{1}{2}})}^{2} = {(8 cos (arcsin (x)))}^{2}

${({(64 - 64 x^{2})}^{\frac{1}{2}})}^{2} = {(8 \cos (\arcsin (x)))}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify

{({(64 - 64 x^{2})}^{\frac{1}{2}})}^{2}

${({(64 - 64 x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in

{({(64 - 64 x^{2})}^{\frac{1}{2}})}^{2}

${({(64 - 64 x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

{(64 - 64 x^{2})}^{\frac{1}{2} \cdot 2} = {(8 cos (arcsin (x)))}^{2}

${(64 - 64 x^{2})}^{\frac{1}{2} \cdot 2} = {(8 \cos (\arcsin (x)))}^{2}$

Step 3.2.1.1.2

Cancel the common factor of

2

$2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(64 - 64 x^{2})}^{\frac{1}{2} \cdot 2} = {(8 \cos (\arcsin (x)))}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(64 - 64 x^{2})}^{1} = {(8 \cos (\arcsin (x)))}^{2}$

Step 3.2.1.2

Simplify.

$64 - 64 x^{2} = {(8 \cos (\arcsin (x)))}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify

${(8 \cos (\arcsin (x)))}^{2}$ .

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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

$(\sqrt{1^{2} - x^{2}}, x)$ ,

$(\sqrt{1^{2} - x^{2}}, 0)$ , and the origin. Then

$\arcsin (x)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through

$(\sqrt{1^{2} - x^{2}}, x)$ . Therefore,

$\cos (\arcsin (x))$ is

$\sqrt{1 - x^{2}}$ .

$64 - 64 x^{2} = {(8 \sqrt{1 - x^{2}})}^{2}$

Step 3.3.1.1.2

Rewrite

$1$ as

$1^{2}$ .

$64 - 64 x^{2} = {(8 \sqrt{1^{2} - x^{2}})}^{2}$

Step 3.3.1.2

Since both terms are perfect squares, factor using the difference of squares formula,

$a^{2} - b^{2} = (a + b) (a - b)$ where

$a = 1$ and

$b = x$ .

$64 - 64 x^{2} = {(8 \sqrt{(1 + x) (1 - x)})}^{2}$

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

$8 \sqrt{(1 + x) (1 - x)}$ .

$64 - 64 x^{2} = 8^{2} {\sqrt{(1 + x) (1 - x)}}^{2}$

Step 3.3.1.3.2

Raise

$8$ to the power of

$2$ .

$64 - 64 x^{2} = 64 {\sqrt{(1 + x) (1 - x)}}^{2}$

Step 3.3.1.3.3

Rewrite

${\sqrt{(1 + x) (1 - x)}}^{2}$ as

$(1 + x) (1 - x)$ .

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Step 3.3.1.3.3.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{(1 + x) (1 - x)}$ as

${((1 + x) (1 - x))}^{\frac{1}{2}}$ .

$64 - 64 x^{2} = 64 {({((1 + x) (1 - x))}^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.3.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$64 - 64 x^{2} = 64 {((1 + x) (1 - x))}^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.3.3

Combine

$\frac{1}{2}$ and

$2$ .

$64 - 64 x^{2} = 64 {((1 + x) (1 - x))}^{\frac{2}{2}}$

Step 3.3.1.3.3.4

Cancel the common factor of

$2$ .

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Step 3.3.1.3.3.4.1

Cancel the common factor.

$64 - 64 x^{2} = 64 {((1 + x) (1 - x))}^{\frac{2}{2}}$

Step 3.3.1.3.3.4.2

Rewrite the expression.

$64 - 64 x^{2} = 64 {((1 + x) (1 - x))}^{1}$

Step 3.3.1.3.3.5

Simplify.

$64 - 64 x^{2} = 64 ((1 + x) (1 - x))$

Step 3.3.1.4

Expand

$(1 + x) (1 - x)$ using the FOIL Method.

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Step 3.3.1.4.1

Apply the distributive property.

$64 - 64 x^{2} = 64 (1 (1 - x) + x (1 - x))$

Step 3.3.1.4.2

Apply the distributive property.

$64 - 64 x^{2} = 64 (1 \cdot 1 + 1 (- x) + x (1 - x))$

Step 3.3.1.4.3

Apply the distributive property.

$64 - 64 x^{2} = 64 (1 \cdot 1 + 1 (- x) + x \cdot 1 + x (- x))$

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

$1$ by

$1$ .

$64 - 64 x^{2} = 64 (1 + 1 (- x) + x \cdot 1 + x (- x))$

Step 3.3.1.5.1.2

Multiply

$- x$ by

$1$ .

$64 - 64 x^{2} = 64 (1 - x + x \cdot 1 + x (- x))$

Step 3.3.1.5.1.3

Multiply

$x$ by

$1$ .

$64 - 64 x^{2} = 64 (1 - x + x + x (- x))$

Step 3.3.1.5.1.4

Rewrite using the commutative property of multiplication.

$64 - 64 x^{2} = 64 (1 - x + x - x \cdot x)$

Step 3.3.1.5.1.5

Multiply

$x$ by

$x$ by adding the exponents.

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Step 3.3.1.5.1.5.1

Move

$x$ .

$64 - 64 x^{2} = 64 (1 - x + x - (x \cdot x))$

Step 3.3.1.5.1.5.2

Multiply

$x$ by

$x$ .

$64 - 64 x^{2} = 64 (1 - x + x - x^{2})$

Step 3.3.1.5.2

Add

$- x$ and

$x$ .

$64 - 64 x^{2} = 64 (1 + 0 - x^{2})$

Step 3.3.1.5.3

Add

$1$ and

$0$ .

$64 - 64 x^{2} = 64 (1 - x^{2})$

Step 3.3.1.6

Apply the distributive property.

$64 - 64 x^{2} = 64 \cdot 1 + 64 (- x^{2})$

Step 3.3.1.7

Multiply.

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Step 3.3.1.7.1

Multiply

$64$ by

$1$ .

$64 - 64 x^{2} = 64 + 64 (- x^{2})$

Step 3.3.1.7.2

Multiply

$- 1$ by

$64$ .

$64 - 64 x^{2} = 64 - 64 x^{2}$

Step 4

Solve for

$x$ .

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Step 4.1

Move all terms containing

$x$ to the left side of the equation.

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Step 4.1.1

Add

$64 x^{2}$ to both sides of the equation.

$64 - 64 x^{2} + 64 x^{2} = 64$

Step 4.1.2

Combine the opposite terms in

$64 - 64 x^{2} + 64 x^{2}$ .

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Step 4.1.2.1

Add

$- 64 x^{2}$ and

$64 x^{2}$ .

$64 + 0 = 64$

Step 4.1.2.2

Add

$64$ and

$0$ .

$64 = 64$

Step 4.2

Since

$64 = 64$ , the equation will always be true for any value of

$x$ .

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$