In the given right triangle, the sides are \[8\] units and \[15\] units long.

\[\theta\]\[8\]\[15\]

**Evaluate the expression without the use of a calculator.**

\[\sec(\theta) + \cosec(\theta)=\] 

To evaluate the expression \(\sec(\theta) + \cosec(\theta)\), we'll first need to determine the necessary trigonometric functions using the given side lengths of the right triangle.

### Step 1: Find the hypotenuse using the Pythagorean theorem.

In a right triangle, the Pythagorean theorem states:

\[
\text{Hypotenuse}^2 = \text{Opposite side}^2 + \text{Adjacent side}^2
\]

Given the sides of the triangle as \(8\) units and \(15\) units, we have:

\[
\text{Hypotenuse}^2 = 8^2 + 15^2 = 64 + 225 = 289
\]

Taking the square root of both sides:

\[
\text{Hypotenuse} = \sqrt{289} = 17
\]

### Step 2: Calculate \(\sec(\theta)\) and \(\cosec(\theta)\).

- **Secant** is the reciprocal of cosine, so:

  \[
  \sec(\theta) = \dfrac{\text{Hypotenuse}}{\text{Adjacent side}} = \dfrac{17}{15}
  \]

- **Cosecant** is the reciprocal of sine, so:

  \[
  \cosec(\theta) = \dfrac{\text{Hypotenuse}}{\text{Opposite side}} = \dfrac{17}{8}
  \]

### Step 3: Add the values of \(\sec(\theta)\) and \(\cosec(\theta)\).

\[
\sec(\theta) + \cosec(\theta) = \dfrac{17}{15} + \dfrac{17}{8}
\]

To add these fractions, find a common denominator:

\[
\text{Common denominator} = 15 \times 8 = 120
\]

Convert each fraction:

\[
\dfrac{17}{15} = \dfrac{17 \times 8}{120} = \dfrac{136}{120}
\]

\[
\dfrac{17}{8} = \dfrac{17 \times 15}{120} = \dfrac{255}{120}
\]

Add the fractions:

\[
\sec(\theta) + \cosec(\theta) = \dfrac{136}{120} + \dfrac{255}{120} = \dfrac{136 + 255}{120} = \dfrac{391}{120}
\]

Thus, the value of the expression is:

\[
\sec(\theta) + \cosec(\theta) = \dfrac{391}{120}
\]

Would you like further details or have any other questions?

Here are some related questions:
1. What are the definitions of secant and cosecant in a right triangle?
2. How can we use the Pythagorean theorem to find missing sides in a right triangle?
3. What is the relationship between trigonometric functions and their reciprocals?
4. How do we add fractions with different denominators?
5. Can you determine \(\sin(\theta)\) and \(\cos(\theta)\) using the same triangle?

**Tip:** When working with trigonometric identities in right triangles, it's essential to identify which sides correspond to the opposite, adjacent, and hypotenuse relative to the given angle.

Learn how to evaluate \(\sec(\theta) + \cosec(\theta)\) in a right triangle with sides 8 units and 15 units. Detailed step-by-step solution using trigonometric ratios and the Pythagorean theorem. Suitable for high school students studying trigonometry.

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