Find the exact value of cos theta in simplest radical form given an angle in standa...
Check the final answer first, then review the worked steps.
Problem
Find the exact value of cos theta in simplest radical form given an angle in standard position whose terminal side passes through the point (-35, 12).
Answer
\(\cos \theta = -\frac{35}{37}\)
Step-by-step solution
- Identify the coordinates: The problem states that the terminal side of angle $\theta$ passes through the point $(-35, 12)$. This means $x = -35$ and $y = 12$.
2. Calculate the distance r: The distance $r$ from the origin to the point $(x, y)$ is calculated using the Pythagorean theorem: $r = \sqrt{x^2 + y^2}$.
Substitute the given values: $r = \sqrt{(-35)^2 + (12)^2}$
$r = \sqrt{1225 + 144}$
$r = \sqrt{1369}$
$r = 37$
3. Determine the cosine value: The cosine of an angle in standard position is defined as $\cos \theta = \frac{x}{r}$.
Substitute the values of $x$ and $r$: $\cos \theta = \frac{-35}{37}$
- Simplify the result: The fraction $\frac{-35}{37}$ is already in its simplest form, and it does not contain any radicals.