If A = 22.5°, what is the value of (10√2 sin 2A − 7√2 cos 2A + 9 tan 2A)?

Concept: sin 45° = cos 45° = √2⁄2 (i.e. 1/√2), and tan 45° = 1 — these are the standard exact trigonometric values at the special angle 45°. Doubling A = 22.5° gives the special angle 2A = 45°, so every trig function in the expression can be replaced by its exact value rather than approximated.

Application: Substitute 2A = 45° into each term and simplify step by step:

1. Given A = 22.5°, so 2A = 2 × 22.5° = 45°.

2. At 2A = 45°: sin 2A = √2⁄2, cos 2A = √2⁄2, and tan 2A = 1.

3. First term: 10√2 sin 2A = 10√2 × (√2⁄2) = 10 × (2⁄2) = 10.

4. Second term: 7√2 cos 2A = 7√2 × (√2⁄2) = 7 × (2⁄2) = 7, so with its minus sign this term contributes −7.

5. Third term: 9 tan 2A = 9 × 1 = 9.

6. Adding the three computed terms in the order given: 10 − 7 + 9 = 12.

Cross-check: Independently check with decimals: √2 ≈ 1.4142, so sin 45° ≈ cos 45° ≈ 0.7071. Then 10 × 1.4142 × 0.7071 ≈ 10.000, −7 × 1.4142 × 0.7071 ≈ −7.000, and 9 × 1 = 9.000. Summing: 10.000 − 7.000 + 9.000 = 12.000, matching the exact computation above to three decimal places.