Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. One of the key concepts in trigonometry is the sum formula, which helps us find the trigonometric functions of the sum of two angles.
Sum Formulas
The sum formulas for sine, cosine, and tangent are essential tools in trigonometry. Let’s break down these formulas and see how they work.
Sine Sum Formula
The sine of the sum of two angles $A$ and $B$ is given by:
$sin(A + B) = sin A cos B + cos A sin B$
Example
Suppose you want to find $sin(45^text{°} + 30^text{°})$. Using the sum formula:
$sin(45^text{°} + 30^text{°}) = sin 45^text{°} cos 30^text{°} + cos 45^text{°} sin 30^text{°}$
We know that:
$sin 45^text{°} = frac{sqrt{2}}{2}$, $cos 30^text{°} = frac{sqrt{3}}{2}$, $cos 45^text{°} = frac{sqrt{2}}{2}$, and $sin 30^text{°} = frac{1}{2}$
So,
$sin(45^text{°} + 30^text{°}) = frac{sqrt{2}}{2} cdot frac{sqrt{3}}{2} + frac{sqrt{2}}{2} cdot frac{1}{2}$
$= frac{sqrt{6}}{4} + frac{sqrt{2}}{4}$
$= frac{sqrt{6} + sqrt{2}}{4}$
Cosine Sum Formula
The cosine of the sum of two angles $A$ and $B$ is given by:
$cos(A + B) = cos A cos B – sin A sin B$
Example
Suppose you want to find $cos(45^text{°} + 30^text{°})$. Using the sum formula:
$cos(45^text{°} + 30^text{°}) = cos 45^text{°} cos 30^text{°} – sin 45^text{°} sin 30^text{°}$
We use the same known values:
$cos 45^text{°} = frac{sqrt{2}}{2}$, $cos 30^text{°} = frac{sqrt{3}}{2}$, $sin 45^text{°} = frac{sqrt{2}}{2}$, and $sin 30^text{°} = frac{1}{2}$
So,
$cos(45^text{°} + 30^text{°}) = frac{sqrt{2}}{2} cdot frac{sqrt{3}}{2} – frac{sqrt{2}}{2} cdot frac{1}{2}$
$= frac{sqrt{6}}{4} – frac{sqrt{2}}{4}$
$= frac{sqrt{6} – sqrt{2}}{4}$
Tangent Sum Formula
The tangent of the sum of two angles $A$ and $B$ is given by:
$tan(A + B) = frac{tan A + tan B}{1 – tan A tan B}$
Example
Suppose you want to find $tan(45^text{°} + 30^text{°})$. Using the sum formula:
$tan(45^text{°} + 30^text{°}) = frac{tan 45^text{°} + tan 30^text{°}}{1 – tan 45^text{°} tan 30^text{°}}$
We know that:
$tan 45^text{°} = 1$ and $tan 30^text{°} = frac{1}{sqrt{3}}$
So,
$tan(45^text{°} + 30^text{°}) = frac{1 + frac{1}{sqrt{3}}}{1 – 1 cdot frac{1}{sqrt{3}}}$
$= frac{1 + frac{1}{sqrt{3}}}{1 – frac{1}{sqrt{3}}}$
$= frac{sqrt{3} + 1}{sqrt{3} – 1}$
By rationalizing the denominator, we get:
$tan(45^text{°} + 30^text{°}) = 2 + sqrt{3}$
Conclusion
The sum formulas for sine, cosine, and tangent are powerful tools in trigonometry. They allow us to find the trigonometric functions of the sum of two angles, making it easier to solve complex problems. Understanding these formulas and how to apply them is essential for anyone studying trigonometry.