Write the trigonometric expression as an algebraic expression in and .
Assume that the variables and represent positive real numbers.

Ask by Ortiz Park. in the United States

Dec 05,2024

Question

Write the trigonometric expression as an algebraic expression in and .
Assume that the variables and represent positive real numbers.

Ask by Ortiz Park. in the United States

Dec 05,2024

UpStudy AI · Accepted Answer

To express the trigonometric expression $$\sin \left(\cos ^{-1} u-\tan ^{-1} v\right)$$ as an algebraic expression in $$u$$ and $$v$$, we can use the trigonometric identities and properties.

Given:
- $$u$$ and $$v$$ are positive real numbers.

We know that:
- $$\cos^{-1} u = \arccos u$$
- $$\tan^{-1} v = \arctan v$$

Using the trigonometric identity $$\sin(\arccos x) = \sqrt{1-x^2}$$ and $$\sin(\arctan x) = \frac{x}{\sqrt{1+x^2}}$$, we can rewrite the expression as:

$$
\sin \left(\cos ^{-1} u-\tan ^{-1} v\right) = \sin \left(\arccos u-\arctan v\right)
$$

Now, we can use the trigonometric identity $$\sin(a-b) = \sin a \cos b - \cos a \sin b$$ to expand the expression further:

$$
\sin \left(\arccos u-\arctan v\right) = \sin(\arccos u) \cos(\arctan v) - \cos(\arccos u) \sin(\arctan v)
$$

Substitute the expressions for $$\sin(\arccos u)$$ and $$\cos(\arctan v)$$ using the identities mentioned earlier:

$$
\sin \left(\arccos u-\arctan v\right) = \sqrt{1-u^2} \cdot \frac{1}{\sqrt{1+v^2}} - \frac{u}{\sqrt{1+u^2}} \cdot \frac{v}{\sqrt{1+v^2}}
$$

Therefore, the trigonometric expression $$\sin \left(\cos ^{-1} u-\tan ^{-1} v\right)$$ can be expressed as the algebraic expression:

$$
\sqrt{1-u^2} \cdot \frac{1}{\sqrt{1+v^2}} - \frac{u}{\sqrt{1+u^2}} \cdot \frac{v}{\sqrt{1+v^2}}
$$
$$ \sqrt{1-u^2} \cdot \frac{1}{\sqrt{1+v^2}} - \frac{u}{\sqrt{1+u^2}} \cdot \frac{v}{\sqrt{1+v^2}} $$

Write the trigonometric expression as an algebraic expression in and .
Assume that the variables and represent positive real numbers.

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Write the trigonometric expression as an algebraic expression in and . Assume that the variables and represent positive real numbers.

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Write the trigonometric expression as an algebraic expression in and .
Assume that the variables and represent positive real numbers.