Given sin 𝛼 =
3
5
𝑖𝑛 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡 𝐼 𝑎𝑛𝑑 cos 𝛽 = −
5
13
𝑖𝑛 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡 𝐼𝐼, find
(a) sin(𝛼 − 𝛽)

To solve for \( \sin(\alpha - \beta) \), we use the following identity for sine of a difference of angles:

\[
\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta
\]

### Given:
- \( \sin \alpha = \frac{3}{5} \) in quadrant I.
- \( \cos \beta = -\frac{5}{13} \) in quadrant II.

### Step 1: Find \( \cos \alpha \)
Since \( \alpha \) is in quadrant I, both \( \sin \alpha \) and \( \cos \alpha \) are positive. We use the Pythagorean identity:

\[
\sin^2 \alpha + \cos^2 \alpha = 1
\]

Substitute \( \sin \alpha = \frac{3}{5} \):

\[
\left( \frac{3}{5} \right)^2 + \cos^2 \alpha = 1
\]
\[
\frac{9}{25} + \cos^2 \alpha = 1
\]
\[
\cos^2 \alpha = 1 - \frac{9}{25} = \frac{16}{25}
\]
\[
\cos \alpha = \frac{4}{5}
\]

### Step 2: Find \( \sin \beta \)
Since \( \beta \) is in quadrant II, \( \sin \beta \) is positive. Again, using the Pythagorean identity:

\[
\sin^2 \beta + \cos^2 \beta = 1
\]

Substitute \( \cos \beta = -\frac{5}{13} \):

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

### Step 3: Substitute into the identity
Now substitute \( \sin \alpha = \frac{3}{5} \), \( \cos \alpha = \frac{4}{5} \), \( \sin \beta = \frac{12}{13} \), and \( \cos \beta = -\frac{5}{13} \) into the sine difference formula:

\[
\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta
\]
\[
\sin(\alpha - \beta) = \left( \frac{3}{5} \right) \left( -\frac{5}{13} \right) - \left( \frac{4}{5} \right) \left( \frac{12}{13} \right)
\]
\[
\sin(\alpha - \beta) = \frac{-15}{65} - \frac{48}{65}
\]
\[
\sin(\alpha - \beta) = \frac{-63}{65}
\]

Thus, the value of \( \sin(\alpha - \beta) \) is:

\[
\boxed{\frac{-63}{65}}
\]

Would you like further clarification or more details on this? Here are 5 related questions to deepen understanding:

1. How do you find \( \cos(\alpha + \beta) \) given the same information?
2. What is the value of \( \tan(\alpha - \beta) \)?
3. How does the quadrant of an angle affect the sign of trigonometric functions?
4. Can you use the sum and difference identity for cosine to find \( \cos(\alpha - \beta) \)?
5. What is the geometric meaning of \( \sin(\alpha - \beta) \)?

**Tip:** When working with trigonometric identities, always check the quadrant of the angle to determine the correct sign of the trigonometric functions.

Given sin(𝛼) = 3/5 in quadrant I and cos(𝛽) = -5/13 in quadrant II, find sin(𝛼 − 𝛽).

Learn how to solve for sin(𝛼 − 𝛽) when given sin(𝛼) = 3/5 in quadrant I and cos(𝛽) = -5/13 in quadrant II. This problem covers key trigonometric identities like the sine difference formula and the Pythagorean identity. Suitable for students in Grades 10-12.

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