Short Answer 
To determine the distance before heading North to a friend’s house, you first establish that the total distance is 10 kilometers and the distance East is 6 kilometers. By applying the Pythagorean theorem, you find that the distance North is 8 kilometers, leading to a sine value of 0.8 for the angle at your friend’s house.
Step 1: Understand the Distances
To determine how far you need to ride before heading North to reach your friend’s house, you first need to grasp the relevant distances. In this scenario:
- The total distance from your house to your friend’s house, termed as c, is 10 kilometers.
- The distance from your friend’s house to a point directly East of it, referred to as a, is 6 kilometers.
- Your goal is to find the distance between this direct East point and where you turn North, recognized as b.
Step 2: Apply the Pythagorean Theorem
Next, you will use the Pythagorean theorem to solve for the unknown distance b. The theorem allows us to relate the sides of a right triangle:
- The formula is c² = a² + b².
- For this example, substituting the known values gives us 10² = 6² + b².
- This simplifies to 100 = 36 + b², leading to b² = 64, which gives b = 8 kilometers.
Step 3: Calculate the Sine of the Angle
Lastly, to find the sine of the angle (Œ∏) at your friend’s house, you can utilize the sine formula, which relates the angle, opposite side, and hypotenuse:
- The sine is calculated as sin θ = opposite/hypotenuse.
- In this case, the opposite side is the distance b = 8 km and the hypotenuse is c = 10 km.
- Thus, sin θ = 8/10 = 0.8 reveals the sine value at that point.