AI Tutor
If you punch sin⁑(75∘)\sin(75^\circ) into a calculator, you get 0.9659258...0.9659258..., an endless decimal that conceals the angle's exact geometry. How can you write this value as a precise radical like 6+24\frac{\sqrt{6}+\sqrt{2}}{4} without any rounding error?
The secret lies in breaking unfamiliar angles down into standard benchmark anglesβ€”specifically 30∘30^\circ (Ο€6\frac{\pi}{6}), 45∘45^\circ (Ο€4\frac{\pi}{4}), and 60∘60^\circ (Ο€3\frac{\pi}{3})β€”whose exact trigonometric coordinates we already know from special right triangles.
What algebraic bridge lets us combine these known ratios to find the sine or cosine of their sum or difference?

The Compound Angle Formulas

A compound angle is an angle formed by the algebraic sum or difference of two or more distinct angles, such as (A+B)(A + B) or (Aβˆ’B)(A - B). In the 2nd century, the astronomer Claudius Ptolemy used geometric chords to calculate angle additions for stellar navigation, laying the foundation for our modern sine and cosine identities.
How do we put these formulas to work on our opening mystery angle, 75∘75^\circ?

Evaluating sin⁑(75∘)\sin(75^\circ)

We start by decomposing 75∘75^\circ into two benchmark angles whose exact ratios are known: 75∘=45∘+30∘75^\circ = 45^\circ + 30^\circ. Applying the sine addition identity gives sin⁑(45∘+30∘)=sin⁑(45∘)cos⁑(30∘)+cos⁑(45∘)sin⁑(30∘)\sin(45^\circ + 30^\circ) = \sin(45^\circ)\cos(30^\circ) + \cos(45^\circ)\sin(30^\circ).
Now substitute the known exact values: (22)(32)+(22)(12)=64+24=6+24\left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}.
Notice why the common shortcut sin⁑(A+B)=sin⁑A+sin⁑B\sin(A+B) = \sin A + \sin B is a fatal misconception: sin⁑(45∘)+sin⁑(30∘)=22+12β‰ˆ1.207\sin(45^\circ) + \sin(30^\circ) = \frac{\sqrt{2}}{2} + \frac{1}{2} \approx 1.207, which is impossible because sine can never exceed 11.
Degrees are straightforward to add, but what happens when angles are written in radians?