To understand the Law of Sines and the Law of Cosines for triangles that are not right-angled, let’s break it down into simple steps.
Law of Sines:
Look at a Triangle: Imagine triangle ABC. It has angles (A), (B), and (C), and the sides opposite these angles are (a), (b), and (c).
Draw a Height: From point A, draw a straight line down to side (BC). This line is called the height and splits the triangle into two smaller right triangles.
Use Trigonometry: In these right triangles, you can use the sine function. For angle (A), we can say: [ \sin A = \frac{h}{a} ] You can do the same for angles (B) and (C).
Find Relationships: Rearranging gives: [ h = a \cdot \sin A ] And for angle (B), [ b = \frac{h}{\sin B} ]
Connect Everything: By putting these together, you get the Law of Sines: [ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} ]
Law of Cosines:
Use Triangle ABC Again: Start with triangle ABC and remember what cosine means.
Set Up the Cosine Rule: You want to find a link between the sides and angles of the triangle.
Make a Right Triangle: Use the sides of the triangle and look at angle (C). You will find that: [ c^2 = a^2 + b^2 - 2ab \cdot \cos C ]
General Rule: You can do the same for angles (A) and (B), which gives you the complete Law of Cosines: [ c^2 = a^2 + b^2 - 2ab \cdot \cos C ]
Both of these laws are super helpful! They let you find unknown sides or angles in any triangle, not just right-angled ones!
To understand the Law of Sines and the Law of Cosines for triangles that are not right-angled, let’s break it down into simple steps.
Law of Sines:
Look at a Triangle: Imagine triangle ABC. It has angles (A), (B), and (C), and the sides opposite these angles are (a), (b), and (c).
Draw a Height: From point A, draw a straight line down to side (BC). This line is called the height and splits the triangle into two smaller right triangles.
Use Trigonometry: In these right triangles, you can use the sine function. For angle (A), we can say: [ \sin A = \frac{h}{a} ] You can do the same for angles (B) and (C).
Find Relationships: Rearranging gives: [ h = a \cdot \sin A ] And for angle (B), [ b = \frac{h}{\sin B} ]
Connect Everything: By putting these together, you get the Law of Sines: [ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} ]
Law of Cosines:
Use Triangle ABC Again: Start with triangle ABC and remember what cosine means.
Set Up the Cosine Rule: You want to find a link between the sides and angles of the triangle.
Make a Right Triangle: Use the sides of the triangle and look at angle (C). You will find that: [ c^2 = a^2 + b^2 - 2ab \cdot \cos C ]
General Rule: You can do the same for angles (A) and (B), which gives you the complete Law of Cosines: [ c^2 = a^2 + b^2 - 2ab \cdot \cos C ]
Both of these laws are super helpful! They let you find unknown sides or angles in any triangle, not just right-angled ones!