Derive and use addition and double-angle formulas

Try

Find sin(π/12) exactly by writing π/12=π/4−π/6. Subtracting the two sine values is not valid. Instead, represent a rotation through one angle followed by another. A rotated point’s new vertical coordinate depends on both its old horizontal and vertical coordinates.

Predict

Predict the signs in a cosine addition formula by choosing a=b=π/2. The sum angle is π, whose cosine is −1. Since both individual cosines are zero and both sines are one, the sine-product term must be subtracted, not added.

Experiment

A point (x,y) rotated counterclockwise through angle a becomes (x cos a−y sin a, x sin a+y cos a). Apply this mapping to (cos b,sin b). Its new angle is a+b, giving the two coordinate identities:

• cos(a+b)=cos a cos b−sin a sin b.
• sin(a+b)=sin a cos b+cos a sin b.

Check the mapping on (1,0) and (0,1) for a=π/2. They become (0,1) and (−1,0), consistent with a quarter-turn. This verifies the orientation of the signs in a concrete case.

Understand

Replacing b by −b gives difference formulas because cosine is even and sine is odd. Thus sin(a−b)=sin a cos b−cos a sin b. For π/12, substitute the exact π/4 and π/6 values to obtain (√6−√2)/4. The resulting radical expression is exact; rounding each intermediate value would unnecessarily lose precision.

Setting a=b=θ yields sin2θ=2sinθcosθ and cos2θ=cos²θ−sin²θ. Using sin²θ+cos²θ=1 gives alternative forms cos2θ=1−2sin²θ=2cos²θ−1. Choose the form matching the information available. If only sin θ is known, the second form determines cos2θ without needing the sign of cos θ.

Dividing the sine addition formula by the cosine addition formula gives tan(a+b)=(tan a+tan b)/(1−tan a tan b), where both individual tangents exist and the new denominator is nonzero. At a=b=π/4, that denominator vanishes because the sum angle is π/2, where tangent is undefined. This is a domain condition, not a reason to assign infinity as a function value.

Addition formulas also justify repeated rotations in vector and complex-number calculations. The same mechanism appears later when multiplying two complex numbers in polar form: their arguments add because the product acts as a rotation and scaling.

Apply

In the derivation practice, show how each double-angle form follows without division, so no extra angles are excluded. In the tangent problem, state every denominator condition. A useful numerical check chooses angles with exact values, but the proof must still explain the general formula.

Continue

Now use identities to solve equations. Transformations may produce several branches, and periodicity means a correct principal value may represent only a small part of the full solution set.