Questions: Use the appropriate reciprocal identity to find the exact value of sec θ for the given value of cos θ. cos θ = 8/9 sec θ = (Type an integer or a fraction.)

$Use the appropriate reciprocal identity to find the exact value of sec θ for the given value of cos θ. cos θ = 8/9 sec θ = (Type an integer or a fraction.)$

Transcript text: Use the appropriate reciprocal identity to find the exact value of $\sec \theta$ for the given value of $\cos \theta$. \[ \cos \theta=\frac{8}{9} \] $\boldsymbol{\operatorname { s e c }} \theta=$ $\square$ (Type an integer or a fraction.)

Solution

Solution Steps

To find the exact value of $\sec \theta$ given $\cos \theta = \frac{8}{9}$, we use the reciprocal identity for secant, which states that $\sec \theta = \frac{1}{\cos \theta}$. By substituting the given value of $\cos \theta$ into this identity, we can calculate $\sec \theta$.

Step 1: Identify the Reciprocal Identity

To find $\sec \theta$ given $\cos \theta = \frac{8}{9}$, we use the reciprocal identity for secant: \[ \sec \theta = \frac{1}{\cos \theta} \]

Step 2: Substitute the Given Value

Substitute the given value of $\cos \theta$ into the reciprocal identity: \[ \sec \theta = \frac{1}{\frac{8}{9}} \]

Step 3: Simplify the Expression

Simplify the expression by taking the reciprocal of $\frac{8}{9}$: \[ \sec \theta = \frac{9}{8} \]