The Law of Sines in 3D

The Law of Sines is a fundamental principle in trigonometry that relates the ratios of the sides of a triangle to the angles opposite them. This law is typically introduced in the context of planar triangles, that is, triangles that lie entirely in a two-dimensional plane. However, the Law of Sines can also be applied to triangles that exist in three-dimensional space.

To understand how this works, we first need to review the Law of Sines as it applies to planar triangles. In a planar triangle with sides of length a, b, and c opposite angles A, B, and C respectively, the Law of Sines states that:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

This formula allows us to solve for unknown sides or angles of a triangle when we know some of the other sides and angles. For example, if we know the lengths of sides a and b and the measure of angle A, we can use the Law of Sines to find the length of side c:

$\frac{c}{\sin C} = \frac{a}{\sin A} = \frac{b}{\sin B}$

$\Rightarrow c = \frac{\sin C}{\sin A}a = \frac{\sin C}{\sin B}b$

Now let's consider a triangle that exists in three-dimensional space. We can still apply the Law of Sines to this triangle, but we need to be a bit more careful about what we mean by "sides" and "angles". In particular, we need to be clear about which angles are opposite which sides.

In the case of a 3D triangle, we can define the "sides" to be the lengths of the line segments that connect the vertices of the triangle. These line segments are also known as the "edges" of the triangle. We can then define the "angles" to be the angles between these edges. Specifically, if we label the edges of the triangle as a, b, and c, and the angles between them as A, B, and C respectively, then we can apply the Law of Sines to this triangle as follows:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Notice that in this formula, the side lengths a, b, and c are now three-dimensional vectors rather than simple scalars. The angles A, B, and C are still just the measures of the angles between these vectors.

So how do we apply this formula to solve problems involving 3D triangles? Let's look at an example.

Suppose we have a triangle with vertices at (1, 2, 3), (4, 5, 6), and (7, 8, 11). We want to find the measure of angle A, which is opposite the edge connecting the first two vertices.

To solve this problem, we first need to find the lengths of the other two edges of the triangle. We can do this using the distance formula:

$b = \sqrt{(4-1)^2 + (5-2)^2 + (6-3)^2} = \sqrt{27}$

$c = \sqrt{(7-1)^2 + (8-2)^2 + (11-3)^2} = \sqrt{174}$

Now we can use the Law of Sines to find the measure of angle A:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

$\Rightarrow \frac{a}{\sin A} = \frac{\sqrt{27}}{\sin B} = \frac{\sqrt{174}}{\sin C}$

We can solve for sin A as follows:

$\frac{a}{\sqrt{27}} = \frac{\sin A}{\sin B} = \frac{\sqrt{174}}{c}\sin A$

$\Rightarrow \sin A = \frac{a}{\sqrt{27}}\cdot\frac{c}{\sqrt{174}} = \frac{ac}{\sqrt{26\cdot 27}}$

Now we can use the inverse sine function to find the measure of angle A:

$A = \sin^{-1}\left(\frac{ac}{\sqrt{26\cdot 27}}\right)$

Plugging in the values of a and c that we found earlier, we get:

$A = \sin^{-1}\left(\frac{\sqrt{27}\cdot\sqrt{174}}{\sqrt{26\cdot 27}}\right) = \sin^{-1}\left(\sqrt{\frac{174}{26}}\right) \approx 58.86^\circ$

So we have found that angle A is approximately 58.86 degrees.

In conclusion, the Law of Sines is a powerful tool that can be applied to triangles in both two and three dimensions. By understanding how to interpret sides and angles in a 3D triangle, we can use the Law of Sines to solve a wide range of problems involving triangles in three-dimensional space.

Links

3D正弦定理[JA]