Two sides of triangle are 6cm and 7cm and the angle between them is 40°. Find the third side?
Answers
The overall steps are:
Draw a triangle consistent with the given information, labeling relevant information
Determine which formulas make sense in the situation (Area of entire triangle based on two fixed-length sides, and trig relationships of right triangles for the variable height)
Relate any unknown variables (height) back to the variable (θ) which corresponds to the only given rate (dθdt)
Do some substitutions into a "main" formula (the area formula) so that you can anticipate using the given rate
Differentiate and use the given rate to find the rate you are aiming for (dAdt)
Let's write down the information given formally:
dθdt=0.07 rad/s
Then you have two fixed-length sides and an angle between them. The third length is a variable value, but it is technically an irrelevant length. What we want is dAdt. There is no indication that this is a right triangle, however, so let's start by assuming that it's not at the moment.
A theoretically consistent triangle is:

Keep in mind that this is not proportionally representative of the true triangle. The area of this can be found most easily with:
A=B⋅h2
where our base is of course 6. What is h, though? If we draw a dividing line vertically from the apex down to the base, we automatically have a right triangle on the left side of the overall triangle, regardless of the length of side x:

Now we do have a right triangle. Notice, however, that our area formula has h but not θ, and we only know dθdt. So, we need to represent h in terms of an angle. Knowing that the only known side on the left-hand right triangle is the 7-lengthed side:
sinθ=h7
7sinθ=h
So far, we have:
dθdt=0.07 rad/s (1)
A=Bh2 (2)
7sinθ=h (3)
So, we can plug (3) into (2), differentiate (2) and implicitly acquire dθdt, and plug (1) into (2) to solve for dAdt, our goal:
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