https://file.aiquickdraw.com/mathbot/file/1726195059117hpupfr3m.jpg

To solve the trigonometric values of angle \( A \) given that \( a = 3 \) and \( c = 5 \), we'll first need to determine the length of side \( b \) using the Pythagorean theorem:

\[
c^2 = a^2 + b^2
\]

Substituting the given values:

\[
5^2 = 3^2 + b^2
\]

\[
25 = 9 + b^2
\]

\[
b^2 = 16
\]

\[
b = 4
\]

Now that we have all the side lengths, we can calculate the trigonometric functions for angle \( A \):

1. **\(\sin(A)\)**: 

   \[
   \sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c} = \frac{3}{5} = 0.6
   \]

2. **\(\cos(A)\)**: 

   \[
   \cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{c} = \frac{4}{5} = 0.8
   \]

3. **\(\tan(A)\)**: 

   \[
   \tan(A) = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{b} = \frac{3}{4} = 0.8
   \]

4. **\(\sec(A)\)**:

   \[
   \sec(A) = \frac{1}{\cos(A)} = \frac{1}{0.8} = 1.3
   \]

5. **\(\csc(A)\)**: 

   \[
   \csc(A) = \frac{1}{\sin(A)} = \frac{1}{0.6} = 1.7
   \]

6. **\(\cot(A)\)**: 

   \[
   \cot(A) = \frac{1}{\tan(A)} = \frac{1}{0.8} = 1.3
   \]

**Summary:**

- \(\sin(A) = 0.6\)
- \(\cos(A) = 0.8\)
- \(\tan(A) = 0.8\)
- \(\sec(A) = 1.3\)
- \(\csc(A) = 1.7\)
- \(\cot(A) = 1.3\)

Would you like any more details on this solution or have any questions?

Here are some related questions you might explore:

1. How do you calculate trigonometric values for other angles in a triangle?
2. What happens to trigonometric values as the angle changes?
3. How can these trigonometric functions be used in real-world applications?
4. How do you determine the trigonometric values of non-right triangles?
5. What are inverse trigonometric functions, and how are they used?

**Tip:** Always ensure the triangle is a right triangle when using these basic trigonometric ratios!

Suppose a = 3 and c = 5 in a right triangle. Find the values of sin(A), cos(A), tan(A), sec(A), csc(A), and cot(A), rounded to one decimal place.

Calculate the trigonometric ratios sin(A), cos(A), tan(A), sec(A), csc(A), and cot(A) for a right triangle where a = 3 and c = 5. Includes detailed step-by-step solution using the Pythagorean theorem. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation