Two very important topics in trigonometry are the Laws of Cosine and Sine. These laws help with finding angles and side lengths of non-right angle triangles. We already have our tools and equations for right angle triangles, but not much for any other triangle. Let's dive in!
The law of cosine follows the form of:
[ 1 ]
$$ a^2 = b^2 + c^2 - 2bc \cos{\alpha} $$
[ 2 ]
$$ b^2 = a^2 + c^2 - 2ac \cos{\beta} $$
[ 3 ]
$$ c^2 = a^2 + b^2 - 2ab \cos{\gamma} $$
Image of non-right angle triangle.
Above is an example of a triangle marked with the used symbols from Eq. (1-3). It is helpful when working with similar triangles, like the one in the image, to mark them a, b, c, and to mark the angles $ \alpha $, $ \beta $, $ \gamma $. This way, it is quicker and easier to use the Law of Cosine.
The law of sine follows the form:
[ 4 ]
$$ \frac{a}{\sin{\alpha}} = \frac{a}{\sin{\beta}} = \frac{c}{\sin{\gamma}} $$
We can use the same triangle example from before to model our variables off of. Getting used to both of these powerful formulas will help with many applications such as trigonometry and vector analysis.