trigonometry question: how do you find the area of a triangle when given 2 sides and 1 angle measurement?

Question

ok so I have to find the area of a triangle with the following measurements:

A=58 degrees
b=10 meters
c=21 meters

I have to round the answer to the nearest hundreth

I don't just want an answer I need a step by step walk through so I can understand how it's done and apply it to other problems. Any help is appreciated.

if you draw it out as a triangle and label it properly it's a SAS triangle

Answer 1

ok.. there are a lot of ways to solve it, but prob., the formula you are looking for is

area= b x c x SinA x .5 (I can't do a 'divided by 2' on this machine)

and why does this work.

first off, you have a fixed triangle because you have a basic Side angle Side triangle. That means that you could compute the angles B and C based upon the proportions between the angles and sides and the fact that the remaining two angles must be equal to 122 degrees.

Knowing that you could drop a line from one apex perpendicular to the opposite side and then compute the length of that side by the same proportionment since you would know two of the three angles. knowing the perpendicular (aka the height) would let you compute the area of each of the two smaller triangles on either side.

If you combine the formula for computing those two small triangles and then simplify it you will get the original formula that I typed.

ta da...

And you get bonus credits for asking HOW and not WHAT the answer is.. not to many students have the self esteem necessary to ask it that way.. good for you.

Answer 2

Picture c as the side of the triangle on the ground, b as the left side slanted to the right so it makes a 58° angle with c. The distance from the top of side b down to the ground is the height of the triangle, and it's equal to b sin 58. The area is 1/2 base times height, so that's (1/2)bc sin 58. It works this easily because A = 58° is the angle between b and c. Things could be harder if that were not the case.

so you get (1/2)(10)(21) sin 58 = 89.05 m²

Answer 3

That is not enough information. Is the angle the included (opposite) angle (i.e. is it the angle between the two sides for which you gave lengths)? If so there is a unique answer. If not, there may or may not be.

Remember

SAS proves congruence or uniqueness
SSA does not - at least not always.

If it is the included angle, the formula for area is:

Area = (1/2)bc sin A = (1/2)(10)(21)(sin 58°)
= 105(sin 58°) = 89.04505 meters²

Answer 4

Presumably b & c are the sides adjacent to the angle A. Using c for the base of the triangle, the height is bsinA, and
A = (1/2)cbsinA
A = (1/2)(21 m)(10 m)sin58°
A = (21 m)(5 m)(0.848048)
A = 89.045 m^2

Answer 5

In any triangle, we have
S = 1/2 ab sinC = 1/2 ac sin B = 1/2 bc sinA
thus S here = 1/2 bc sinA = 89.045 m^2

Answer 6

There are 60 minutes is a degree: a' = (a/60)° There are 60 seconds in a minuter, or 60*60 = 3600 seconds in a degree: b'' = (b/3600)° 87°26'3" = (87 + 26/60 + 3/3600)° = 87.434° -------------------- When going clockwise, degrees are negative A full rotation clockwise will be -360° 120 full rotations clockwise = 120 * -360° = -43,200° -------------------- -230° + 360° = 130° -------------------- cos(q) = -12 / √(12²+5²) = -12/13 -------------------- tan = 0 when sin = 0 i.e. when both sides of angle are along x-axis 0°, 180°, 360°, 540° --------------------