Sketch a right triangle corresponding to the trigonometric function of the acute angle 𝜃. Then find the exact values of the other five trigonometric functions of 𝜃.
sin(𝜃) = 
5
13
cos(𝜃)	=	
 
tan(𝜃)	=	
 
csc(𝜃)	=	
 
sec(𝜃)	=	
 
cot(𝜃)	=	
 


We are given that \( \sin(\theta) = \frac{5}{13} \), and we need to find the remaining five trigonometric functions of the acute angle \( \theta \). Let's proceed step by step.

### Step 1: Set up a right triangle based on \( \sin(\theta) \)

The sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse. So we can say:

- Opposite side \( = 5 \)
- Hypotenuse \( = 13 \)

Now, we need to find the adjacent side using the Pythagorean theorem:
\[
\text{(Hypotenuse)}^2 = (\text{Opposite})^2 + (\text{Adjacent})^2
\]
\[
13^2 = 5^2 + (\text{Adjacent})^2
\]
\[
169 = 25 + (\text{Adjacent})^2
\]
\[
(\text{Adjacent})^2 = 144
\]
\[
\text{Adjacent} = \sqrt{144} = 12
\]

### Step 2: Find the remaining trigonometric functions

Now that we know the opposite side is 5, the adjacent side is 12, and the hypotenuse is 13, we can find the other trigonometric functions:

1. **Cosine**:
\[
\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{12}{13}
\]

2. **Tangent**:
\[
\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{5}{12}
\]

3. **Cosecant**:
\[
\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{13}{5}
\]

4. **Secant**:
\[
\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{13}{12}
\]

5. **Cotangent**:
\[
\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{12}{5}
\]

### Summary of all trigonometric functions for \( \theta \):
- \( \sin(\theta) = \frac{5}{13} \)
- \( \cos(\theta) = \frac{12}{13} \)
- \( \tan(\theta) = \frac{5}{12} \)
- \( \csc(\theta) = \frac{13}{5} \)
- \( \sec(\theta) = \frac{13}{12} \)
- \( \cot(\theta) = \frac{12}{5} \)

If you would like, I can create a sketch of this triangle for better visualization.

Would you like more details on any part of this solution?

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**5 related questions:**
1. How does the Pythagorean theorem relate to trigonometric functions?
2. What are the properties of the reciprocal trigonometric functions?
3. How do you find trigonometric functions for angles in non-right triangles?
4. What happens to the trigonometric functions if the angle \( \theta \) is in a different quadrant?
5. How can trigonometric identities be used to simplify expressions involving these functions?

**Tip**: Remember that the reciprocal trigonometric functions (csc, sec, cot) can be found directly from their primary counterparts (sin, cos, tan).

Sketch a right triangle corresponding to the trigonometric function of the acute angle 𝜃. Then find the exact values of the other five trigonometric functions of 𝜃. sin(𝜃) = 5/13

Explore how to find the remaining trigonometric functions of an acute angle 𝜃 when sin(𝜃) = 5/13. Learn the step-by-step process of using the Pythagorean theorem to complete the right triangle and calculate the values for cos(𝜃), tan(𝜃), csc(𝜃), sec(𝜃), and cot(𝜃). Ideal for students in Grades 9-12.

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