High school mathematics; known a [2,0] B [0,2] C [cos θ, sin θ], O as the origin of coordinates. Vector ac * vector BC = - 1 / 3

High school mathematics; known a [2,0] B [0,2] C [cos θ, sin θ], O as the origin of coordinates. Vector ac * vector BC = - 1 / 3

Vector AC = (COS θ - 2, sin θ) BC = (COS θ, sin θ - 2)
Vector ac * BC = cos θ (COS θ - 2) + sin θ (sin θ - 2) = 1-2 (sin θ + cos θ) = - 1 / 3 = = > sin θ + cos θ = 2 / 3 square
1+sin2θ=4/9===>sin2θ=-5/9.

In the plane rectangular coordinate system, O is the coordinate origin, and the vectors a = (2,1), a (1,0), B (COS Θ, t) are known
(1) If the vector AB is parallel to the vector a, and the module of AB is equal to √ 5 times the module of OA, the coordinates of the vector ob are obtained;
(2) If vector AB is parallel to vector a, find y = cos square Θ - cos Θ + T square

The first question is (3,1), the second question is to seek the maximum, the maximum or the minimum

Let t = √ 1 + sin2 θ (1) given sin (π - θ) = 3 / 5 and θ as obtuse angle, find the value of T; (2) given cos (π / 2 - θ) = m and θ as obtuse angle, find the value of T

Let t = √ 1 + sin2 θ = root sign (sin θ + cos θ) ^ 2 = | sin θ + cos θ|
sin（π-θ）=sinθ=3/5,
If θ is an obtuse angle, then cos θ = - 4 / 5
T=|3/5-4/5|=1/5
(2) Cos (π / 2 - θ) = sin θ = m > 0, θ is an obtuse angle, then cos θ = - radical (1-m ^ 2)
T = | m-radical (1-m ^ 2)|

In Cartesian coordinates, a (2,0), B (0,2), C (COS θ, sin θ) are known
(1) If θ is an obtuse angle and sin θ = 3 / 5, find the vector ab · CB
(2) If the vector Ca ⊥ the vector CB, find the value of sin2 θ

(1)
A（2,0）,B(0,2),C(cosθ,sinθ)
Vector AB = (- 2,2), CB = (- cos θ, 2-sin θ)
∵θ is an obtuse angle, and sin θ = 3 / 5,
∴cosθ=-4/5
Ψ vector ab · vector CB
=2cosθ+2(2-sinθ)
=4-8/5-6/5
=6/5
(2)
Vector CA = (2-cos θ, - sin θ)
Vector Ca ⊥ vector CB
The vector Ca ● the vector CB = 0
∴(2-cosθ,-sinθ)●(-cosθ,2-sinθ)
=(2-cosθ)(-cosθ)+(-sinθ)(2-sinθ)
=(cos²θ+sin²θ)-2(sinθ+cosθ)
=1-2(sinθ+cosθ)=0
∴sinθ+cosθ=1/2
Square on both sides:
sin²θ+cos²θ+2sInθcosθ=1/4
∴sin2θ=1/4-1=-3/4