## THE CASE OF SOHCAHTOA JOEChase Arthur and Taylor Tharp

The suspects

In order to find out who SohCahToa Joe is, we need to find his favorite number from the clues we were given.

The first clue was to find the height of the Taj Mahal, given the angle of elevation of 79.9 degrees, and the adjacent length of 100 feet.

Clue given by SohCahToa Joe

In terms of Sine, Cosine, and Tangent, we are given the adjacent length and the included angle, and need to find the opposite length. To do so, we set up the equation using tangent, because we have "a" and we need "o".

Equation we set up

Next, we used the Cross Products Property to isolate the variable SohCahToa Joe gave us, "h".

Use cross products property to multiply

Lastly, we simplified the value, and rounded to the nearest foot, as SohCahToa Joe said to in his clue.

Height of the Taj Mahal, and clue #1

For clue #2, we had to solve for "m", but to do so we had to solve 3 triangles using the Pythagorean Theorem, because they're all right triangles.

All three triangles in clue 2
Pythagorean Theorem formula
Triangle one, we had "b" and "c" values, needing to solve for "a".
Using the value we found in the first triangle, we found the hypotenuse of the 2nd triangle.
For the third triangle, we plugged in the "x" and "y" values to find "m".

We found the second clue, which is m=21

For the third clue, we had to first find all of the angles for the triangle to determine if any were above 50 degrees.

Clue 3
To find angle a, we used the inverse of Sine, because we were finding an angle when we were given "o" and "h".
To find the other angle in the first triangle, angle B, we subtracted the other two angles from 180 (total degrees all three angles in a triangle add up to), and got an angle degree over 50 degrees.

For the next triangle, we were given "o" and "a" lengths, so to find the included angle we used inverse tangent.

To find the last angle, we again subtracted the two angles that we had from 180.

Angle w is larger than 50 degrees

Two of the angles from the two triangles were over 50 degrees, making the first answer C.

For the next one, we first solved for the first value to figure out what the number was.

Use the inverse of cosine to find an angle measure

Next, we plugged in the choices to figure out which measure matched the first one, and option C did match

Option C

Lastly, we drew the triangle the question described.

Triangle described

Then, because we were given the side length for "p", we solved for angle P using the Law of Sines because we were given two sides and an angle measure.

Angle P equals 37.14 degrees

This was option C, and it was true. This concluded that none of the answers were A, so the clue is a=0

For clue 4, we had to find which ramp was the longest, and to do that for the cosine ramp, we used cosine to find the length of both hypotenuses, and then added them together.

Clue 4
Triangle one in Cosine Ramp
Because this second triangle is a special right triangle (30 60 90) we used the formula hypotenuse= 2(short side).
Lastly, we added the two hypotenuse lengths together to find the length of the cosine ramp, which is 13.25ft.

To find the length of the Sine ramp, since they were both special right triangles, we used the formula for the 45 45 90 triangle, hypotenuse= side(square root of 2), and for the 30 60 90 triangle, hypotenuse= 2(short side).

We added the two lengths together to get the length of the Sine ramp.

The Cosine ramp was longer than the Sine ramp, meaning that J=-5

For clue 5, we first had to find how far the plane traveled in 48 seconds, traveling at 853 feet per second.

Clue 5: Find the height of Mount Everest
Distance from plane in the drawing to Mount Everest
We needed to find the distance between the plane's new position and the tip of the mountain to be able to subtract that from the overall distance from the plane to the ground.
Height of the mountain. t=29,029

For the last clue, we needed to find the length of the bridge. To do that, we first found the third missing angle by subtracting the other two from 180, getting 31 degrees. Then, because we were solving for a side and originally given 2 angles, we used the Law of Sines to solve for the length of the bridge.

Clue 6
Using the given side 848 meters and its opposite angle 31, we set it equal to the angle opposite the bridge length and solved for x using cross products property.

This gave us all of the values to solve the formula SohCahToa Joe included in the clues to find his favorite number.

Formula for SohCahToa Joe's favorite number

H=561, M=21, A=0, J=-5, T=29029, L=1315

Plugging those numbers into the formula, we get 3,665 as SohCahToa Joe's favorite number.

Therefore, SohCahToa Joe is... Carlo.

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