關於math sin/cos/tan同probability

1.
1 - cos²θ
= (sin²θ + cos²θ) - cos²θ
= sin²θ + cos²θ - cos²θ
= sin²θ


2.
sin²θ + cos²θ = 1
Hence, sin²θ = 1 - cos²θ

7cosθ - 3sin²θ + 5 = 0
7cosθ - 3(1 - cos²θ) + 5 = 0
7cosθ - 3 + 3cos²θ + 5 = 0
3cos²θ + 7cosθ + 2 = 0
(3cosθ + 1)(cosθ + 2) = 0
cosθ = -1/3 or cosθ = -2 (rejected)
θ = (180 - 70.5)º or θ = (180 + 70.5)º
θ = 109.5º or θ = 250.5º


3.
P(Peter hits) = 1/5
P(Peter misses) = 1 - (1/5) = 4/5
P(John hits) = 1/4
P(John misses) = 1 - (1/4) = 3/4

(a)
P(the target is not hit)
= P(Peter misses twice and Peter misses twice)
= P(Peter misses twice) x P(Peter misses twice)
= (4/5)² x (3/4)²
= (16/25) x (9/16)
= 9/25

(b)
P(the target is hit)
= 1 - P(the target is not hit)
= 1 - (9/25)
= 16/25

(c)
P(only Peter hits)
= P(only Peter hits once) + P(only Peter hits twice)
= (1/5)(4/5)(3/4)² + (1/5)²(3/4)²
= 36/400 + 9/400
= 45/400

P(only John hits)
= P(only John hits once) + P(only John hits twice)
= (4/5)²(1/4)(3/4) + (4/5)²(1/4)²
= 48/400 + 16/400
= 64/400

P(only one hits)
= P(only Peter hits) + P(only John hits)
= (45/400) + (64/400)
= 109/400