Question

Suppose cot(日) = = 2, where

○
12
5
○ 13
ㅇㅇㅇ
13
晉
where n ㅠ
ㅇ픔

Answer 1

Answer: sinθ=-12/13

Explanation:

`\displaystyle\\Given:\ cot\theta=(5)/(12) \ \ \ \ \pi < \theta < (3\pi )/(2) \ \ \ \ Find:\ sin\theta\\\\(cos\theta)/(sin\theta) =(5)/(12)`

Multiply both parts of the equation by 12sinθ:

`12cos\theta=5sin\theta`

Squared both parts of the equation:

`(12cos\theta)^2=(5sin\theta)^2\\\\144cos^2\theta=25sin^2\theta\ \ \ \ (1)\\`

`Find\ sin\theta`

Add 144sin²θ to both parts of the equation (1):

`144cos^2\theta+144sin^2\theta=25sin^2\theta+144sin^2\theta\\\\144(sin^2\theta+cos^2\theta)=169sin^2\theta\\\\144(1)=169sin^2\theta\\\\144=169sin^2\theta\\\\Thus,\ 169sin^2\theta=144`

Divide both parts of the equation by 169:

`\displaystyle\\sin^2\theta=(144)/(169) \\\\sin\theta=б\sqrt{(144)/(169) } \\\\sin\theta=б\sqrt{(12^2)/(13^2) } \\\\sin\theta=б\sqrt{((12)/(13))^2 } \\\\sin\theta=б(12)/(13)`

`\displaystyle\\As, \ \pi < \theta < (3\pi )/(2) \ \ \ \ \Rightarrow\ \ \ \angle\theta\ is \ in\ the\ 3rd\ quadrant\\\\ So \ sin\theta\ takes \ negative \ values\\\\sin\theta=-(12)/(13)`