Trigonometry Examples

\sin (\arccos (x))

$\sin (\arccos (x))$

Step 1

Write the expression using exponents.

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Step 1.1

Draw a triangle in the plane with vertices

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

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

(x, 0)

$(x, 0)$ , and the origin. Then

\arccos (x)

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

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

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

\sin (\arccos (x))

$\sin (\arccos (x))$ is

\sqrt{1 - x^{2}}

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

\sqrt{1 - x^{2}}

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

Step 1.2

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

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

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

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

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

Step 2

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

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

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

a = 1

$a = 1$ and

b = x

$b = x$ .

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

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