compute sec(-t) from the given value cos(t)=-5/8

To find \(\sec(-t)\), we need to follow these steps:

### 1. Relationship between \(\sec(-t)\) and \(\cos(-t)\):
The secant function is the reciprocal of the cosine function, so:
\[
\sec(-t) = \frac{1}{\cos(-t)}
\]

### 2. Use the Even-Odd Property of Cosine:
The cosine function is an even function, which means:
\[
\cos(-t) = \cos(t)
\]
So, \(\sec(-t) = \frac{1}{\cos(t)}\).

### 3. Substitute the Given Value:
Given \(\cos(t) = -\frac{5}{8}\), substitute this into the equation for \(\sec(-t)\):
\[
\sec(-t) = \frac{1}{-\frac{5}{8}} = -\frac{8}{5}
\]

### Final Answer:
\[
\sec(-t) = -\frac{8}{5}
\]

Would you like more details, or do you have any questions?

Here are some related questions to explore:

1. What is the value of \(\sec(t)\) if \(\cos(t) = -\frac{5}{8}\)?
2. How does the secant function behave for different values of \(t\)?
3. What is the relationship between \(\sec(t)\) and \(\cos(t)\)?
4. How can you compute \(\csc(-t)\) given \(\sin(t)\)?
5. What are the properties of even and odd functions in trigonometry?

**Tip:** Remember that even functions satisfy \(f(-x) = f(x)\), while odd functions satisfy \(f(-x) = -f(x)\). This helps simplify trigonometric calculations involving negative angles.

Compute sec(-t) from the given value cos(t)=-5/8

Discover how to calculate sec(-t) using the given value of cos(t) = -5/8. This guide explores key trigonometric concepts, including the even function property and the reciprocal relationship between secant and cosine. Ideal for students in Grades 10-12.

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