Use this calculator to find one missing side or angle in a triangle.
Knowing how to use the law of sines is a fundamental skill in trigonometry! Here’s a quick review:
What is the Law of Sines?
The law of sines established a relationship between the sides and angles of any triangle. It states that the ratio of the length of a side to the sine of its opposite angle is the same for all three sides.
This is the law of sines formula:
$$\dfrac{a}{\sin(A)}=\dfrac{b}{\sin(B)}=\dfrac{c}{\sin(C)}$$
The lowercase letters represent the side lengths, and the uppercase letters represent the angles that are opposite those sides.
Example: Finding a Missing Side
Let’s find the missing length of side $b$ in a triangle where we know the following:
- $a=10$
- $\alpha=40°$
- $\beta=60°$
We only need to use parts of the formula that involve our known values and the side we want to find, which looks like this:
$$\dfrac{a}{\sin(\alpha)}=\dfrac{b}{\sin(\beta)}$$
To get $b$ by itself, we need to multiply both sides of the equation by $\sin(\beta)$:
$$b=\dfrac{a\times \sin(\beta)}{\sin(\alpha)}$$
Next, substitute the given measurements into the rearranged formula:
$$b=\dfrac{10\times \sin(60°)}{\sin(40°)}$$
Finally, find the sine values and solve the equation:
$$b \approx \dfrac{10\times 0.866}{0.643}\approx 13.47$$
Therefore, the length of Side $b$ is around 13.47.
