FORMULABANK

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

Example:

The sine rule is a trigonometric formula used to find unknown sides or angles in any triangle (not necessarily right-angled). The formula connects the sides and angles in the triangle.

The sine rule is:

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

How to use the sine rule:

1. Determine what you know: You need to know at least two sides and one angle, or two angles and one side.
2. Write the sine rule for the relevant sides and angles: For example, if you know two sides $a$ and $b$ and one angle $A$, you can find $B$ by using:

$\frac{a}{\mathrm{sin(A)}}=\frac{b}{\mathrm{sin(B)}}$

3. Solve the equation: Use algebra to isolate the unknown value (either an angle or a side). If you're finding an angle, you'll often need to take the arcsin (inverse sine) on both sides.

Example:

Let's say you have a triangle where:

• $\mathrm{a\; =\; 7}$
• $\mathrm{b\; =\; 10}$
• $\mathrm{A\; =\; 30°}$

You want to find the angle $B$.

1. Use the sine rule:

$\frac{7}{\mathrm{sin(30°)}}=\frac{10}{\mathrm{sin(B)}}$

2. Solve for $B$:

$\frac{7}{0.5}=\frac{10}{\mathrm{sin(B)}}$ $14=\frac{10}{\mathrm{sin(B)}}$ $\mathrm{sin(B)}=\frac{10}{14}=0.714$ $B=\mathrm{arcsin(0.714)}\approx \mathrm{45.6°}$

So the angle $B$ is approximately 45.6 degrees.