Prove the identity cos (x+(y-pi/2)) = sin (x+y)?

Question

1. bob is driving along a straight and level road straight toward a mountain. At some point on his trip he measures the angle of elevation to the top of the mountain and finds it to be 21 degrees 31'. He then drives 1 mile (1 mile = 5280 ft) more and measures the angle of elevation to be 33 degrees 20'. Find the height of the mountain to the nearest foot.

2. At an altitude of 2800 feet, the engine on a small plane fails. What angle of glide is needed to reach an airport runway that is 3 miles away by land?

3. From a boat on the river below a dam, the angle of elevation to the top of the dam is 23 degrees 26'. If the dam is 2911 feet above the level of the river, how far is the boat from the base of the dam?

4. Prove the identity cos (x+(y-pi/2)) = sin (x+y)

please help best answer gets 10 points!

Raymond · Accepted Answer

In general
cos(90 - X) = sin(X)

The cosine function is symmetrical about 0.
This means that cos(-X) = cos(X)

combine the two effects and you get

sin[X] = cos[(90-X)] = cos[-(90-X)] = cos[(X - 90)]

The above is in degrees.  For the equivalent in radians, just replace the 90 with pi/2

For your identity, just replace X with (x+y)

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3.
Tan(X) = opposite/adjacent
X is the angle of elevation, opposite is the height of the dam, adjacent is the horizontal distance from the boat to the dam.

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2. Tan(X) = opposite/adjacent
X is the angle of glide, opposite is the altitude (or vertical distance to go down), adjacent is the horizontal distance to the airport.  Make sure that both are in the same units (for example, both in feet).

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1.
This is a difference of tangents problem.

At first stop, Tan(X1) = opposite / adjacent = h/d
X1 is the angle of elevation (21 31), h is the height of the mountain (unknown) and d is the distance to the mountain (also unknown)

At second stop,
Tan(X2) = opposite / adjacent = h / (d-1)
d-1 because the distance is now one less mile (or (d - 5280) if you are doing this in feet)

You now have two equations with two unknowns.  Solve the system.
Remember: h is the same h for both equations (and d is the same d for both equations).

The simplest way is to turn each equation into this form:

h = something / something

Since h is the same in both equations, just make the two right sides equal
(this will give you one equation with only one unknown).

Remember that if you do it this way, d is the distance at the first stop.

Once you have d, go back to the first equation to find h.

? · Answer

by way of fact the question is contained in this style of Pi, this we could us be attentive to that we would desire to paintings in radian style. So set your calculator to the RAD putting Use cos(A +/- B) = cosA*cosB -/+ sinA*sinB Now LHS (left-hand side) = cos (x - [Pi/2]) = cos x * cos [Pi/2] + sin x * sin [Pi/2] all of us be attentive to cos [Pi/2] = 0 and that sin [Pi/2] = a million consequently the equation will become --> cos x * 0 + sin x * a million = 1sin x = sin x = RHS (top-hand side)

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Prove the identity cos (x+(y-pi/2)) = sin (x+y)?

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