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Hey it's a me again @drifter1!
Today we continue with the small series about Trigonometry.
I suggest reading the first and second parts before getting into this one.
Today we will cover how we solve non-right-angled Triangles
So, without further ado, let's get straight into it!
When speaking of Right Triangles, because an angle is already known (the right angle of 90°), only two more values are needed, which can be either angles or sides, in order to solve the whole triangle. More specifically, the solution can be found using Pythagoras's Theorem and the basic trigonometric functions (sin, cos, tan) quite easily.
But, what about non-right triangles, or any general triangle, so to speak?
In general, to solve a triangle, you need to know:
Let's consider the following general triangle:
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The Law of Sines states that:
It's also common to turn the fractions upside-down:
The Law of Sines can be used to:
For example, knowing angle A and sides a and b, we can calculate angle B. Similarly, knowing the angles B and C, and the side c, we can calculate side b.
In the case of SSA problems (two sides with the angle not between them), the remaining angle can give:
The Law of Cosines is a extended version of Pythagoras's Theorem.
Solving for side c in a general triangle, the term of -2abcos(C) has to be added on the left side, or mathematically:
The cosine of an angle of 90° or π/2 radians is 0, which explains why Pythagoras's Theorem works on Right-Angled Triangles.
The Law of Cosines can be used to:
It's easy to just remember the c^2 = ... case and switch out the letters!To calculate the angles use one of the following:
Similarly, one can either solve for cos(X) = ... from the correct x^2 = ... form, or remember the cos(C) = ... caseRemember that the cosine is positive in Quadrant A and negative in Quadrant B, which means that there are no equal values, and so only one solution comes out of the Cosine Rule.
So, now that the Laws are out of the way, let's get into each of the previously mentioned triangle cases!
AAA triangles (all angles known) are basically impossible to solve. Using Trigonometry its only possible to find the shape, but not the actual size. The length of at least one side is necessary in order to continue...
AAS and ASA triangles (two angles and one side which is between or not between the angles) are solved as follows:
SAS triangles (two sides and the angle between them) are solved as follows:
SSA triangles (two sides and an angle not between them) are solved as follows:
SSS triangles (all sides known) are solved as follows:
Because two angles are known, using the rule of 180°, the remaining angle B is easily calculated to be:
Applying the Law of Sines, knowing side a, and angles A and B, its easy to calculate side b:
Applying the Law of Sines, again, its now also possible to calculate side c, using either side/angle pair:
All the sides and angles have now been calculated. Triangle solved!
Because two sides are known, as well as the angle between them, using the Law of Cosines, its easy to calculate the remaining side c:
Next apply the Law of Sines for the smaller of the two angles, which in this case is a, and thus calculate angle A:
Lastly, from the 180° rule, the remaining angle B is:
Two sides are known, but not the angle between them, which means that the Law of Cosines doesn't work in this case. Thus, it's necessary to apply the Law of Sines in order to calculate one of the other angles.
Because sides a and c, as well as the angle A are known, using The Law of Sines angle C is calculated to be:
The remaining angle B can be calculated using the rule of 180°, giving:
Applying the Law of Sinues once more, side b can also be calculated:
And so the triangle is Isosceles!
In this case, only the Law of Cosines works. The Law is applied twice in order to get two of the angles. The last angle can be calculated using the rule of 180°.
Let's first calculate angle C:
Let's next calculate angle A:
Using the rule of 180°, angle B is:
And this is it! The last triangle has been solved!
Mathematical equations used in this article, where made using quicklatex.
