
If  
cos
θ
=
8
17
  and  
θ
  is in Quadrant  
IV
 , find  
csc
θ
  and  
tan
θ
 .

Given that \( \cos \theta = \frac{8}{17} \) and \( \theta \) is in Quadrant IV, we need to find \( \csc \theta \) and \( \tan \theta \).

### Step 1: Determine \( \sin \theta \)

We can use the Pythagorean identity:
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
Substituting the given value for \( \cos \theta \):
\[
\sin^2 \theta + \left(\frac{8}{17}\right)^2 = 1
\]
\[
\sin^2 \theta + \frac{64}{289} = 1
\]
\[
\sin^2 \theta = 1 - \frac{64}{289}
\]
\[
\sin^2 \theta = \frac{289}{289} - \frac{64}{289} = \frac{225}{289}
\]
\[
\sin \theta = \pm \frac{\sqrt{225}}{17} = \pm \frac{15}{17}
\]

Since \( \theta \) is in Quadrant IV, \( \sin \theta \) is negative:
\[
\sin \theta = -\frac{15}{17}
\]

### Step 2: Calculate \( \csc \theta \)

The cosecant is the reciprocal of sine:
\[
\csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{15}{17}} = -\frac{17}{15}
\]

### Step 3: Calculate \( \tan \theta \)

The tangent is the ratio of sine to cosine:
\[
\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{15}{17}}{\frac{8}{17}} = -\frac{15}{8}
\]

### Final Answers:
- \( \csc \theta = -\frac{17}{15} \)
- \( \tan \theta = -\frac{15}{8} \)

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How do you find the values of other trigonometric functions given one trigonometric function?
2. Why is \( \sin \theta \) negative in Quadrant IV?
3. What is the significance of the Pythagorean identity in trigonometry?
4. How does the quadrant affect the sign of trigonometric functions?
5. Can we determine the angle \( \theta \) from \( \cos \theta = \frac{8}{17} \) without a calculator?

**Tip:** Remember that the signs of trigonometric functions depend on the quadrant in which the angle lies.

If cos θ = 8/17 and θ is in Quadrant IV, find csc θ and tan θ.

Learn how to calculate csc θ and tan θ given cos θ = 8/17 with θ in Quadrant IV. This problem covers key trigonometric concepts, including the Pythagorean identity and the relationship between trigonometric functions. Suitable for students in Grades 10-12.

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