To find the value of $\theta$ such that $\sec(\theta) = \frac{a}{b}$ using cofunction identities, follow these steps:

Step 1: Recall the cofunction identity

The cofunction identity relates the secant function to the cosecant function:

$\sec(\theta) = \csc(90^\circ - \theta)$

This means that if $\sec(\theta) = \frac{a}{b}$, then $\csc(90^\circ - \theta) = \frac{a}{b}$.

Step 2: Set the cofunction equal to $\frac{a}{b}$

We are given that $\csc(16^\circ) = \frac{a}{b}$, so we can compare it to the cofunction identity:

$\csc(16^\circ) = \csc(90^\circ - \theta)$

Step 3: Solve for $\theta$

Since the cosecant functions are equal, we can equate the angles:

$16^\circ = 90^\circ - \theta$

Now solve for $\theta$:

$\theta = 90^\circ - 16^\circ = 74^\circ$

Final Answer:

The value of $\theta$ such that $\sec(\theta) = \frac{a}{b}$ is $\theta = 74^\circ$.

Would you like more details, or have any questions?

Here are 5 related questions:

1. What are the other cofunction identities involving trigonometric functions?
2. How can you derive the secant and cosecant functions from sine and cosine?
3. How does the secant function behave as an angle approaches 90 degrees?
4. Can cofunction identities be applied to any angle or just specific ones?
5. How does knowing one trigonometric function help in finding others?

Tip: Cofunction identities are useful for transforming one trigonometric function into another, especially when dealing with complementary angles.