Derive the law of sines.
Understand the Problem
The question is asking us to derive the law of sines, which involves demonstrating the relationship between the lengths of the sides of a triangle and the sines of its angles. This typically includes using a triangle and applying trigonometric principles.
Answer
$$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $$
Answer for screen readers
The Law of Sines is given by the formula: $$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $$
Steps to Solve
- Draw the Triangle
Start by drawing triangle $ABC$ with sides $a$, $b$, and $c$ opposite angles $A$, $B$, and $C$, respectively. This visual representation will help us understand the relationships we want to establish.
- Height of the Triangle
Drop a perpendicular from point $A$ to side $BC$, denoting the foot of the perpendicular as point $D$. This height divides the triangle into two right triangles: $ABD$ and $ACD$.
- Using Trigonometric Ratios
In triangle $ABD$, we can express the height $AD$ in terms of side $b$ and angle $C$: $$ AD = b \sin C $$
In triangle $ACD$, similarly, we can express the height $AD$ in terms of side $c$ and angle $B$: $$ AD = c \sin B $$
- Setting Heights Equal
Since both expressions equal $AD$, we can set them equal to each other: $$ b \sin C = c \sin B $$
- Deriving the General Formula
Now, we can express the relationship involving sides and angles as follows: $$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $$
- Final Expression of the Law of Sines
This leads us to the Law of Sines: $$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $$
The Law of Sines is given by the formula: $$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $$
More Information
The Law of Sines is a fundamental relation in trigonometry that establishes a relationship between the lengths of the sides of a triangle and the sines of its angles. It can be used to solve triangles when two angles and one side are known, or two sides and a non-included angle.
Tips
- Confusing properties: Students often confuse the Law of Sines with the Law of Cosines. Itâ€™s important to remember that the Law of Sines is specifically for angle-side relationships.
- Forgetting to include all parts: Sometimes, students forget to include all three ratios in their final expressions. Ensure to express it for each side.