For positive acute angles AA and B, commaB, it is known that sine, A, equals, start fraction, 5, divided by, 13, end fractionsinA= 
13
5
​	
  and tangent, B, equals, start fraction, 40, divided by, 9, end fraction, .tanB= 
9
40
​	
 . Find the value of sine, left parenthesis, A, plus, B, right parenthesissin(A+B) in simplest form.

We are tasked with finding \( \sin(A + B) \), given that:

\[
\sin(A) = \frac{5}{13} \quad \text{and} \quad \tan(B) = \frac{40}{9}
\]

To solve for \( \sin(A + B) \), we use the sum identity for sine:

\[
\sin(A + B) = \sin(A) \cos(B) + \cos(A) \sin(B)
\]

### Step 1: Find \( \cos(A) \)
Since we know \( \sin(A) = \frac{5}{13} \) and \( \sin^2(A) + \cos^2(A) = 1 \), we can solve for \( \cos(A) \):

\[
\sin^2(A) + \cos^2(A) = 1
\]
\[
\left( \frac{5}{13} \right)^2 + \cos^2(A) = 1
\]
\[
\frac{25}{169} + \cos^2(A) = 1
\]
\[
\cos^2(A) = 1 - \frac{25}{169} = \frac{144}{169}
\]
\[
\cos(A) = \frac{12}{13}
\]

### Step 2: Find \( \sin(B) \) and \( \cos(B) \)
We are given that \( \tan(B) = \frac{40}{9} \). Using the identity \( \tan(B) = \frac{\sin(B)}{\cos(B)} \), we can construct a right triangle where the opposite side is 40, the adjacent side is 9, and use the Pythagorean theorem to find the hypotenuse:

\[
\text{Hypotenuse} = \sqrt{40^2 + 9^2} = \sqrt{1600 + 81} = \sqrt{1681} = 41
\]

Thus, we have:

\[
\sin(B) = \frac{40}{41}, \quad \cos(B) = \frac{9}{41}
\]

### Step 3: Apply the sum identity
Now that we know \( \sin(A) = \frac{5}{13} \), \( \cos(A) = \frac{12}{13} \), \( \sin(B) = \frac{40}{41} \), and \( \cos(B) = \frac{9}{41} \), we can substitute into the sum identity:

\[
\sin(A + B) = \sin(A) \cos(B) + \cos(A) \sin(B)
\]
\[
\sin(A + B) = \left( \frac{5}{13} \times \frac{9}{41} \right) + \left( \frac{12}{13} \times \frac{40}{41} \right)
\]
\[
\sin(A + B) = \frac{45}{533} + \frac{480}{533}
\]
\[
\sin(A + B) = \frac{525}{533}
\]

Thus, the value of \( \sin(A + B) \) in its simplest form is:

\[
\boxed{\frac{525}{533}}
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you derive the cosine addition formula?
2. What is the difference between sine and cosine rules in trigonometry?
3. How can we use tangent values to find other trigonometric functions?
4. What is the significance of the Pythagorean identity in trigonometry?
5. How do you determine trigonometric values for angles in different quadrants?

### Tip:
Remember that knowing one trigonometric function of an angle allows you to find the others using identities like the Pythagorean identity or tangent formulas.

For positive acute angles A and B, it is known that sin(A) = 5/13 and tan(B) = 40/9. Find the value of sin(A + B) in simplest form.

Learn how to calculate sin(A + B) for acute angles A and B, given sin(A) = 5/13 and tan(B) = 40/9. This problem covers trigonometric identities like the sum of angles identity and the Pythagorean identity. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation