AI Tutor
What happens if you try to distribute sine across addition, like sin⁡(30∘+60∘)=sin⁡(30∘)+sin⁡(60∘)\sin(30^\circ + 60^\circ) = \sin(30^\circ) + \sin(60^\circ)?
Let's check the values: sin⁡(90∘)=1\sin(90^\circ) = 1, but sin⁡(30∘)+sin⁡(60∘)=12+32≈1.366\sin(30^\circ) + \sin(60^\circ) = \frac{1}{2} + \frac{\sqrt{3}}{2} \approx 1.366 — the math breaks completely.
To find the sine of a sum, we use the sine addition formula: sin⁡(a+b)=sin⁡(a)cos⁡(b)+cos⁡(a)sin⁡(b)\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b).
In the 2nd century CE, the astronomer Claudius Ptolemy developed the earliest equivalent of this identity in Alexandria to calculate chord lengths and map planetary orbits.
How can this formula unlock exact values for angles that don't appear on standard reference tables?

Exact Values for Compound Angles

A compound angle is an angle created by adding or subtracting two or more angles, such as 75∘=45∘+30∘75^\circ = 45^\circ + 30^\circ.
Because we know the exact values for 45∘45^\circ and 30∘30^\circ, we can evaluate sin⁡(75∘)\sin(75^\circ) without using a decimal approximation from a calculator.
Carrying out the arithmetic gives sin⁡(75∘)=6+24\sin(75^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4}, which is approximately 0.96590.9659.
What if you aren't given the angle measures at all, but only their trigonometric ratios and quadrants?

Evaluating with Unknown Angles