Answer
In a right triangle, the **secant** of an angle is the ratio of the hypotenuse to the adjacent side. So, \( \sec(\theta) = \frac{\text{Hypotenuse}}{\text{Adjacent Side}} \).
Solution
In the context of a right triangle, the **secant function** (denoted as **sec**) relates one of the non-right angles to the lengths of specific sides of the triangle. Here's how it's defined:
### Definition:
**Secant of an angle** in a right triangle is the ratio of the length of the **hypotenuse** to the length of the **adjacent side** relative to that angle.
### Mathematically:
\[
\sec(\theta) = \frac{\text{Hypotenuse}}{\text{Adjacent Side}}
\]
### Components Explained:
- **Hypotenuse:** The longest side of the right triangle, opposite the right angle.
- **Adjacent Side:** The side that forms the given angle (other than the hypotenuse).
### Illustration:
Consider a right triangle with angle \( \theta \):
```
|
|\
| \
| \
| \
| \ Hypotenuse
| \
|θ \
|_______\
Adjacent
```
- **Angle \( \theta \):** One of the non-right angles in the triangle.
- **Adjacent Side:** The side adjacent to angle \( \theta \) (not the hypotenuse).
- **Hypotenuse:** The side opposite the right angle.
### Example:
Suppose in a right triangle, the adjacent side to angle \( \theta \) is **3 units**, and the hypotenuse is **5 units**. Then:
\[
\sec(\theta) = \frac{5}{3} \approx 1.6667
\]
### Relationship to Other Trigonometric Functions:
- **Secant is the reciprocal of the cosine function:**
\[
\sec(\theta) = \frac{1}{\cos(\theta)}
\]
Since cosine is defined as the ratio of the adjacent side to the hypotenuse:
\[
\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}
\]
Taking the reciprocal gives the secant:
\[
\sec(\theta) = \frac{\text{Hypotenuse}}{\text{Adjacent}}
\]
### Important Notes:
- The secant function is **undefined** for angles where the adjacent side is **zero**, which corresponds to angles where the cosine is zero (e.g., \( 90^\circ \) and \( 270^\circ \) in the unit circle).
- In a right triangle, \( \sec(\theta) \) will always be **greater than or equal to 1** or **less than or equal to -1**, depending on the angle, because the hypotenuse is always at least as long as the adjacent side.
By understanding the secant function in terms of the sides of a right triangle, you can easily relate it to geometric representations and solve various trigonometric problems involving right-angled triangles.
Reviewed and approved by the UpStudy tutoring team
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