Trigonometry Examples

$\sin (\arccos (1 - x))$

Step 1

Write the expression using exponents.

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Step 1.1

Draw a triangle in the plane with vertices $(1 - x, \sqrt{1^{2} - {(1 - x)}^{2}})$ , $(1 - x, 0)$ , and the origin. Then $\arccos (1 - x)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1 - x, \sqrt{1^{2} - {(1 - x)}^{2}})$ . Therefore, $\sin (\arccos (1 - x))$ is $\sqrt{1 - {(1 - x)}^{2}}$ .

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

Step 1.2

Rewrite $1$ as $1^{2}$ .

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

Step 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 $b = 1 - x$ .

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

Step 3

Simplify.

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Step 3.1

Add $1$ and $1$ .

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

Step 3.2

Apply the distributive property.

$\sqrt{(2 - x) (1 - 1 \cdot 1 - - x)}$

Step 3.3

Multiply $- 1$ by $1$ .

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

Step 3.4

Multiply $- - x$ .

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Step 3.4.1

Multiply $- 1$ by $- 1$ .

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

Step 3.4.2

Multiply $x$ by $1$ .

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

Step 3.5

Subtract $1$ from $1$ .

$\sqrt{(2 - x) (0 + x)}$

Step 3.6

Add $0$ and $x$ .

$\sqrt{(2 - x) x}$

Step 4

Reorder factors in $\sqrt{(2 - x) x}$ .

$\sqrt{x (2 - x)}$