To find the values of the trigonometric functions of $\theta$ given that $\csc(\theta) = 4$ and $\theta$ is in Quadrant I, let's go through each step carefully.

Step-by-Step Solution

1. Understand the given information:

   • $\csc(\theta) = 4$
   • $\theta$ is in Quadrant I.

2. Recall the definition of cosecant (csc): $\csc(\theta) = \frac{1}{\sin(\theta)}$

   Given $\csc(\theta) = 4$, we can write: $\frac{1}{\sin(\theta)} = 4$

3. Solve for $\sin(\theta)$: $\sin(\theta) = \frac{1}{4}$

4. Use the Pythagorean identity to find $\cos(\theta)$: The Pythagorean identity states: $\sin^2(\theta) + \cos^2(\theta) = 1$

   Substituting $\sin(\theta) = \frac{1}{4}$: $\left( \frac{1}{4} \right)^2 + \cos^2(\theta) = 1$ $\frac{1}{16} + \cos^2(\theta) = 1$ $\cos^2(\theta) = 1 - \frac{1}{16}$ $\cos^2(\theta) = \frac{16}{16} - \frac{1}{16}$ $\cos^2(\theta) = \frac{15}{16}$

5. Solve for $\cos(\theta)$: Since $\theta$ is in Quadrant I, both sine and cosine are positive: $\cos(\theta) = \sqrt{\frac{15}{16}} = \frac{\sqrt{15}}{4}$

6. Find the other trigonometric functions:

   • Tangent (tan): $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{1}{4}}{\frac{\sqrt{15}}{4}} = \frac{1}{\sqrt{15}} = \frac{\sqrt{15}}{15}$

   • Cotangent (cot): $\cot(\theta) = \frac{1}{\tan(\theta)} = \sqrt{15}$

   • Secant (sec): $\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{4}{\sqrt{15}} = \frac{4\sqrt{15}}{15}$

Summary of Trigonometric Functions

• $\sin(\theta) = \frac{1}{4}$
• $\cos(\theta) = \frac{\sqrt{15}}{4}$
• $\tan(\theta) = \frac{\sqrt{15}}{15}$
• $\csc(\theta) = 4$
• $\sec(\theta) = \frac{4\sqrt{15}}{15}$
• $\cot(\theta) = \sqrt{15}$

Would you like more details on any of the steps, or do you have any questions?

Related Questions:

1. What is the value of $\theta$ if $\sin(\theta) = \frac{1}{4}$ in Quadrant I?
2. How do you find the trigonometric functions if given $\cos(\theta)$ instead?
3. What happens to the values of trigonometric functions in different quadrants?
4. How can you verify the trigonometric identities using these values?
5. What are the reciprocal identities of the trigonometric functions?

Tip: Remember that trigonometric functions have different signs depending on the quadrant in which the angle lies.