Question

Can someone help me find the answer but with the final answer being in radians

Answer 1

(1)
`-(63)/(65)`

(2) 1.318 rad

Explanation:

1.

π ≤ x ≤ 3π/2 → x at Quadran 3 → x-value negative & y-value negative

3π/2 ≤ y ≤ 2π → y at Quadran 4 → x-value positive & y-value negative

∠x :

sin x = -3/5

y : r = -3 : 5

x = √(r² - y²)

= √(5² - (-3)²)

= -4

x : y : r = -4 : -3 : 5

cos x = -4/5

∠y :

sec y = 13/12

cos y = 12/13

x : r = 12 : 13

y = √(r² - x²)

= √(13² - 12²)

= -5

x : y : r = 12 : -5 : 13

sin y = -5/13

cos(x+y) = cos x · cos - sin x · sin y

=
`-(4)/(5) *(12)/(13) -(-(3)/(5) )*(-(5)/(13) )`

=
`-(48)/(65) -(15)/(65)`

=
`-(63)/(65)`

2.

13 - 22cos x - 8sin²x = 0 (assuming incomplete question → the equation = 0)

13 - 22cos x - 8(1 - cos²x) = 0

8cos²x - 22cos x + 5 = 0

(4cos x - 1)(2cos x - 5) = 0

cos x₁ =
`(1)/(4)`

cos x₂ =
`(5)/(2)` (not valid, as -1 ≤ cos x ≤ 1)

cos x =
`(1)/(4)`

x = 1.318 rad