Part A Your GPS shows that your friend’s house is 10.0 km away (Figure 2). But there is a big hill between your houses and you don’t want to bike there directly. You know your friend’s street is 6.0 km north of your street. How far do you have to ride before turning north to get to your friend’s house?
8 km
Part B
Referring to the diagram in Part A, what is the sine of the angle
θ at the location of the friend's house?

Part A

You have to ride 8.0 km before turning north to get to your friend’s house.

Part B

The sine of the angle  θ at the location of the friend's house is 0.8

Explanation:

The remaining part of the question which is an image is attached below

Explanation:

Part A

To determine how far you will ride ride before turning north,

From the diagram, that is the distance of your street.

Let the distance of your street be

and the distance of your friend's street be

and let the displacement between your friends house and your house be

The relation in the diagram shows a right angle triangle.

The sides of the right angle triangle are represented as and .

To find , which is the distance of your street,

From Pythagorean theorem, 'The square of hypotenuse is the sum of squares of the other two sides'

That is,

is the hypotenuse, which is the displacement between your friends house and your house,

Hence,

is adjacent, which is the distance of your friends street

then,

and is the opposite, which is the distance of your house

From Pythagoras theorem, we can then write that,

Then,

Hence, you have to ride 8.0 km before turning north to get to your friend’s house.

Part B

To find the sine of the angle  θ at the location of the friend's house,

In the diagram, the sine of the angle  θ is given by

Hence,

Then,

Hence, the sine of the angle  θ at the location of the friend's house is 0.8