graphically explain the sum identities in trigonometry
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Trigonometric sum identities are fundamental tools in mathematics that allow us to find exact values of trigonometric functions for angles that are sums of known angles. The two main sum identities are sine of A plus B equals sine A cosine B plus cosine A sine B, and cosine of A plus B equals cosine A cosine B minus sine A sine B. These identities can be visualized on the unit circle where we can see how two angles A and B combine to form their sum.
To derive the cosine sum identity geometrically, we set up a coordinate plane with a unit circle and identify three key points. Point P1 is at angle A with coordinates cosine A comma sine A. Point P2 is at angle A plus B with coordinates cosine of A plus B comma sine of A plus B. Point P3 is at angle negative B with coordinates cosine negative B comma sine negative B. These three points will form the foundation for our geometric proof using the distance formula.
Now we apply the distance formula to connect our geometric setup to algebraic manipulation. The key insight is that the distance between P1 and P2 equals the distance between P3 and the point (1,0). We write this as the squared distance from P1 to P2 equals the squared distance from P3 to (1,0). Expanding the left side gives us cosine of A plus B minus cosine A squared plus sine of A plus B minus sine A squared. Expanding the right side gives us 1 minus cosine negative B squared plus 0 minus sine negative B squared, which simplifies to 1 minus cosine B squared plus sine squared B.
Now we complete the algebraic derivation of the cosine sum identity. Expanding the left side of our equation gives us cosine squared of A plus B minus 2 cosine of A plus B cosine A plus cosine squared A, plus sine squared of A plus B minus 2 sine of A plus B sine A plus sine squared A. The right side expands to 1 minus 2 cosine B plus cosine squared B plus sine squared B. Using the Pythagorean identity, we simplify to get 2 minus 2 cosine of A plus B cosine A minus 2 sine of A plus B sine A equals 2 minus 2 cosine B. This gives us our final identity: cosine of A plus B equals cosine A cosine B minus sine A sine B. We can verify this with A equals 30 degrees and B equals 60 degrees, where cosine of 90 degrees equals 0, which matches our formula calculation.
To derive the sine sum identity, we use the co-function relationship between sine and cosine. The key insight is that sine of theta equals cosine of 90 degrees minus theta. Applying this to sine of A plus B, we get cosine of 90 degrees minus A plus B, which equals cosine of 90 degrees minus A minus B. Using the cosine difference identity, this becomes cosine of 90 degrees minus A times cosine B plus sine of 90 degrees minus A times sine B. Since cosine of 90 degrees minus A equals sine A and sine of 90 degrees minus A equals cosine A, we get our final identity: sine of A plus B equals sine A cosine B plus cosine A sine B. The unit circle visualization shows this 90-degree phase shift relationship between sine and cosine functions.