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Trigonometry — §4.2.1, 4.2.2 31 Trigonometry Identities ! sin2 x + cos2 x = 1 ! sin(x + y ) = sin x cos y + cos x sin y Remember key values of sin, cos. ! cos(x − y ) = cos x cos y − sin x sin y Example. Determine the value of cot(π/24). (p. 233) π π π cos 24 2 cos2 24 1 + cos 12 1 + cos( π3 − π4 ) = . π = π π π π = π sin 24 2 sin 24 cos 24 sin 12 sin( 3 − 4 ) Applying the identities, 1 √ 2 + 4 + 46 √ √ 6 2 − 4 4 √ √ √ √ = 2 + 2 + 3 + 6. Trigonometry — §4.2.1, 4.2.2 32 Trigonometry Identities Example. Find all acute angles x satisfying the equation (p. 231) 2 sin x cos 40◦ = sin(x + 20◦ ). Solution. Find a solution that works. Then apply identity, solve for x. 2 sin x cos 40◦ = sin(x + 20◦ ) = sin x cos 20◦ + sin 20◦ cos x. tan x = sin 20◦ /(2 cos 40◦ − cos 20◦ ). For 0◦ ≤ x < 90◦ , tan x = C can only have one solution. Idea: You may have to make use of e i θ = cos θ + i sin θ. Extension: e i (nx) = (e ix )n , so cos nx + i sin nx = (cos x + i sin x)n . Example. Let f (x) = a + b cos 2x + c sin 5x + d cos 8x for a, b, c, d ∈ R. Then we can write f (x) as the real part of a + be 2ix − ice 5ix + de 8ix . Discrete Geometry — §4.1.2, 4.1.6 Discrete Geometry Idea: Draw good pictures; choose good coordinates. Example. Inscribe in a circle a trapezoid T w/one side as diameter and a triangle % with sides parallel to the sides of the trapezoid. Prove that T and % have the same area. (p. 208) Solution. Define T by coordinates (1, 0), (−1, 0), (b, a), (−b, a). Find the coordinates of %: they are (0, 1), (b, −a), and (−b, −a). Idea: Points in the plane can be represented as complex numbers. Example. Let ABC and BCD be two equilateral triangles, and line passing through D passes through AC at M and through AB at N. (p. 210) Prove that the angle between BM and CN is π3 . √ √ Solution. Choose A = i 3, B = −1, C = 1, and D = −i 3. Notice DBN is similar to DCM ⇒ can find the coordinates of M c−n = te i π/3 and N: if |MC | = 2t, then |BN| = 2t . Calculate b−m 33