We are given a right-angled triangle $ABC$, where:

• $\angle B = 90^\circ$
• $AB = 6$ units
• $BC = 8$ units.

We need to find the value of $\sin A \cos C + \cos A \sin C$.

Step 1: Apply the Pythagorean Theorem to find $AC$

Since the triangle is right-angled at $B$, we can use the Pythagorean theorem:

$AC^2 = AB^2 + BC^2$

Substituting the values:

$AC^2 = 6^2 + 8^2 = 36 + 64 = 100$

$AC = \sqrt{100} = 10 \text{ units}.$

Step 2: Find the trigonometric values for $A$ and $C$

• For $\sin A$ and $\cos A$:

  Using the triangle, $\sin A$ is the opposite side over the hypotenuse:

  $\sin A = \frac{BC}{AC} = \frac{8}{10} = 0.8$

  $\cos A$ is the adjacent side over the hypotenuse:

  $\cos A = \frac{AB}{AC} = \frac{6}{10} = 0.6$

• For $\sin C$ and $\cos C$:

  Since $C = 90^\circ - A$ (complementary angles), we have:

  $\sin C = \cos A = 0.6$

  $\cos C = \sin A = 0.8$

Step 3: Calculate $\sin A \cos C + \cos A \sin C$

The expression $\sin A \cos C + \cos A \sin C$ simplifies to $\sin(A + C)$.

Since $A + C = 90^\circ$:

$\sin 90^\circ = 1.$

Thus, the value of $\sin A \cos C + \cos A \sin C = 1$.

Would you like more details or have any further questions?

Here are 5 related questions to explore further:

1. How would you calculate the value of $\tan A$ and $\tan C$?
2. How do complementary angles relate to sine and cosine values?
3. What is the significance of the angle sum identity in trigonometry?
4. How would you apply this method to solve for unknown angles in different types of triangles?
5. Can you derive other trigonometric identities from this setup?

Tip: Always remember that the sum of angles in a triangle is 180°, which is especially useful in right triangles for relating complementary angles.