If the point to be reached is  blocks north and  blocks east, then we may envision a right triangle with sides  and , and hypotenuse . The Pythagorean Theorem tells us that , so we plug in and find that: 

Thus, 

Now, SOHCAHTOA tells us that , so we know that:

Thus, our cosine is approximately .

For the function

the period is equal to

or in this case

which reduces to .

For the function

the period is equal to

or in this case

which reduces to .

For the function

the period is equal to

or in this case

which reduces to .

For a trigonometric function , the period  is equal to . So, for , .

For a trigonometric function , the period  is equal to . So, for , .

For a trigonometric function , the period  is equal to . So, for , .

Multiply out the quadratic equation to get cosΘ² – 2cosΘsinΘ + sinΘ²  

Then use the following trig identities to simplify the expression:

sin2Θ = 2sinΘcosΘ

sinΘ² + cosΘ² = 1 

1 – sin2Θ is the correct answer for (cosΘ – sinΘ)²  

1 + sin2Θ is the correct answer for (cosΘ + sinΘ)² 

The range of a cosine wave is altered by the coefficient placed in front of the base equation. So, if you have , this means that the highest point on the wave will be at  and the lowest at ; however, if you then begin to shift the equation vertically by adding values, as in, , then you need to account for said shift. This would make the minimum value to be  and the maximum value to be .

For our question, the range of values covers . This range is accomplished by having either  or  as your coefficient. ( merely flips the equation over the -axis. The range "spread" remains the same.) We need to make the upper value to be  instead of . To do this, you will need to subtract , or , from . This requires an downward shift of .  An example of performing a shift like this is:

Among the possible answers, the one that works is:

The  parameter does not matter, as it only alters the frequency of the function.

The range of a cosine wave is altered by the coefficient placed in front of the base equation. So, if you have , this means that the highest point on the wave will be at  and the lowest at ; however, if you then begin to shift the equation vertically by adding values, as in, , then you need to account for said shift. This would make the minimum value to be  and the maximum value to be .  

For our question, the range of values covers . This range is accomplished by having either  or  as your coefficient. ( merely flips the equation over the -axis. The range "spread" remains the same.) We need to make the upper value to be  instead of . This requires an upward shift of . An example of performing a shift like this is:

Among the possible answers, the one that works is: