A point on the terminal side of an angle in standard position is given. Find the exact values of , and .

Ask by Wagner Gross. in the United States

Nov 01,2024

Question

A point on the terminal side of an angle $$ 	heta $$ in standard position is given. Find the exact values of $$ \sin 	heta, \cos 	heta $$, and $$ 	an 	heta $$. $$ (4,-4) $$

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To find the exact values of $$ \sin \theta $$, $$ \cos \theta $$, and $$ \tan \theta $$ for the point $$ (4, -4) $$ on the terminal side of the angle $$ \theta $$, we first need to determine the radius $$ r $$ from the origin to the point.

The radius $$ r $$ is calculated using the formula:
$$
r = \sqrt{x^2 + y^2}
$$
where $$ x = 4 $$ and $$ y = -4 $$.

Calculating $$ r $$:
$$
r = \sqrt{4^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}
$$

Next, we can find the values of $$ \sin \theta $$, $$ \cos \theta $$, and $$ \tan \theta $$ using the definitions:
$$
\sin \theta = \frac{y}{r}, \quad \cos \theta = \frac{x}{r}, \quad \tan \theta = \frac{y}{x}
$$

Substituting the values of $$ x $$, $$ y $$, and $$ r $$:
1. For $$ \sin \theta $$:
$$
\sin \theta = \frac{-4}{4\sqrt{2}} = \frac{-1}{\sqrt{2}} = -\frac{\sqrt{2}}{2}
$$

2. For $$ \cos \theta $$:
$$
\cos \theta = \frac{4}{4\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}
$$

3. For $$ \tan \theta $$:
$$
\tan \theta = \frac{-4}{4} = -1
$$

Thus, the exact values are:
$$
\sin \theta = -\frac{\sqrt{2}}{2}, \quad \cos \theta = \frac{\sqrt{2}}{2}, \quad \tan \theta = -1
$$
$$ \sin \theta = -\frac{\sqrt{2}}{2}, \quad \cos \theta = \frac{\sqrt{2}}{2}, \quad \tan \theta = -1 $$

A point on the terminal side of an angle in standard position is given. Find the exact values of , and .

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