Fundamental trigonometric identities are the building blocks for all trigonometric proofs. The most important is the Pythagorean identity: sine squared theta plus cosine squared theta equals one. We also have quotient identities showing tangent as sine over cosine, and cotangent as cosine over sine. Finally, reciprocal identities define secant, cosecant, and cotangent in terms of their reciprocal functions. These identities can be visualized using the unit circle, where key angles like 30, 45, and 60 degrees demonstrate these fundamental relationships.
Now let's prove the Pythagorean identity rigorously using geometric principles. We start with a point P on the unit circle with coordinates cosine theta and sine theta. Using the distance formula, the distance from the origin to point P is the square root of cosine squared theta plus sine squared theta. Since P lies on the unit circle, this distance must equal the radius, which is one. Squaring both sides gives us the fundamental Pythagorean identity: cosine squared theta plus sine squared theta equals one. From this basic identity, we can derive extended forms by dividing through by cosine squared or sine squared to get the secant and cosecant variations.
Sum and difference formulas are essential tools for calculating trigonometric values of compound angles. The sine of A plus B equals sine A cosine B plus cosine A sine B. The cosine of A plus B equals cosine A cosine B minus sine A sine B. These formulas can be proven geometrically using coordinate rotation and vector addition on the unit circle. The difference formulas are derived by substituting negative B for B in the sum formulas. For tangent, the sum formula becomes tangent A plus tangent B divided by one minus tangent A tangent B. These formulas allow us to find exact values for angles that are sums or differences of known angles.
Double angle formulas are special cases of sum formulas where both angles are equal. When we set A equals B equals theta in the sum formulas, we get the double angle formulas. Sine of two theta equals two sine theta cosine theta. For cosine of two theta, we have three equivalent forms: cosine squared theta minus sine squared theta, two cosine squared theta minus one, and one minus two sine squared theta. These different forms are obtained by using the Pythagorean identity to substitute for sine squared or cosine squared terms. The tangent double angle formula is two tangent theta divided by one minus tangent squared theta. These formulas are essential for simplifying expressions and solving equations involving double angles.
Half angle formulas are derived by manipulating double angle formulas to solve for half angles. Starting with the double angle formula for cosine, we can rearrange to get cosine squared of theta over two equals one plus cosine theta over two. Similarly, sine squared of theta over two equals one minus cosine theta over two. Taking square roots gives us the half angle formulas, with plus or minus signs depending on the quadrant. For tangent of theta over two, we have two equivalent forms: one minus cosine theta over sine theta, and sine theta over one plus cosine theta. These formulas are particularly useful for integration and solving trigonometric equations involving half angles.