The Sine Rule
The sin, cos, and tan functions can only be used for right angled triangle, here we will derive the sine rule which can be used for any triangle.
All triangle can be divided into two right angled triangles by selecting a corner and then drawing a perpendicular line to the opposite side as shown below. This is the starting point for the derivation of the sine rule.
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The Cosine Rule
Can be used for all triangles, not just right angled triangles. The cosine rule formula is derived by dividing a scalene triangle into two right angled triangles.
Introduction to the Unit Circle
The unit circle which is a circle with radius 1 is the fundamental concept behind trigonometry.
Derivation of the sine rule.
Derivation of the sine rule.
The angles are labelled with capital letters and the length of the sides are labelled with lowercase letter.
The perpendicular from C to the line AB, divides the triangle ABC into two right angled triangles.
The triangle has a height h.
From the right angled triangle ADC we get: $$ sinA=\frac{h}{b} $$ multiplying both sides by b, we get: \(h=bsinA \)
From the right angled triangle DCB we get: $$ sinB=\frac{h}{a} $$ multiply both sides by a, we get: \(h=asinB \), then comparing the two formulas for h we get: $$ asinB=bsinA $$ and from this we get the sine rule: $$ \frac{sinA}{a}=\frac{sinB}{b} $$
The perpendicular line could have been drawn from B or C to derive the full sine rule formula: $$ \frac{sinA}{a}=\frac{sinB}{b}=\frac{sinC}{c} $$
Exercise: Calculate length of side
Solve the question on paper before seeing the solution.
The Solution
We will use this "version" of the sine rule:
carefully putting in the values and using a calculator, final answer:
Discussion
The Sine rule can be used for any triangle not just right angled triangles. Given two angles and one side you can calculate the other sides. Given one angle and two sides you can calculate the other angles and sides.
The are questions were were only one angle is specified, and two lengths are given, the other length is unknown; in some cases there can be two possible triangles, this is often known as the "ambiguous case of the sine rule" this will be discussed in a future post.