An acute angle, [tex]\theta[/tex], is in a right triangle such that [tex]\cos \theta = \frac{15}{17}[/tex]. What is the value of [tex]\csc \theta[/tex]?

A. [tex]\frac{17}{15}[/tex]
B. [tex]\frac{17}{8}[/tex]
C. [tex]\frac{8}{15}[/tex]
D. [tex]\frac{8}{17}[/tex]



Answer :

Certainly! Let's solve the problem step by step.

We start with the given information that \(\cos \theta = \frac{15}{17}\).

### Step 1: Recall the Pythagorean Identity

The Pythagorean Identity states:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]

### Step 2: Substitute \(\cos \theta\) into the identity

We have \(\cos \theta = \frac{15}{17}\). Therefore,

[tex]\[ \cos^2 \theta = \left(\frac{15}{17}\right)^2 = \frac{225}{289} \][/tex]

### Step 3: Solve for \(\sin^2 \theta\)

Using the identity:

[tex]\[ \sin^2 \theta = 1 - \cos^2 \theta \][/tex]

Substitute \(\cos^2 \theta\):

[tex]\[ \sin^2 \theta = 1 - \frac{225}{289} \][/tex]

To subtract these fractions, we use a common denominator of 289:

[tex]\[ \sin^2 \theta = \frac{289}{289} - \frac{225}{289} = \frac{64}{289} \][/tex]

### Step 4: Solve for \(\sin \theta\)

Since \(\theta\) is an acute angle, \(\sin \theta\) will be positive:

[tex]\[ \sin \theta = \sqrt{\frac{64}{289}} = \frac{8}{17} \][/tex]

### Step 5: Determine \(\csc \theta\)

The cosecant of \(\theta\) is the reciprocal of the sine of \(\theta\):

[tex]\[ \csc \theta = \frac{1}{\sin \theta} \][/tex]

Substitute \(\sin \theta\):

[tex]\[ \csc \theta = \frac{1}{\frac{8}{17}} = \frac{17}{8} \][/tex]

Thus, the value of \(\csc \theta\) is \(\frac{17}{8}\).

The correct answer is:
[tex]\[ \boxed{\frac{17}{8}} \][/tex]