Find the values of \(\cos \Theta\) and \(\tan \Theta\), given that \(\sin \Theta = \frac{8}{9}\) and \(\Theta\) is in Quadrant I.

0\ \ \ \Rightarrow\ \ \ cos \alpha = \frac{ \sqrt{17} }{9} \\\\"> sin^2 \alpha +cos^2 \alpha =1\ \ \ and\ \ \ tan \alpha = \frac{\big{sin \alpha }}{\big{cos \alpha }} \\\\sin \alpha = \frac{8}{9}\\\\ \Rightarrow\ \ \ (\frac{8}{9})^2+cos^2 \alpha =1\ \ \ \Rightarrow\ \ \ cos^2 \alpha =1- \frac{64}{81} \ \ \ \Rightarrow\ \ \ cos^2 \alpha = \frac{17}{81} \\\\ \alpha \ \in\ (0^0;90^0)\ \ \ \Rightarrow\ \ \ cos \alpha >0\ \ \ \Rightarrow\ \ \ cos \alpha = \frac{ \sqrt{17} }{9} \\\\
t an α = 9 8 ​ : 9 17 ​ ​ = 9 8 ​ ⋅ 17 ​ 9 ​ = 17 ​ 8 ​ = 17 ​ ⋅ 17 ​ 8 ⋅ 17 ​ ​ = 17 8 ⋅ 17 ​ ​

Answered by kate200468 | 2024-06-10

The values are cos Θ = 9 17 ​ ​ and tan Θ = 17 8 17 ​ ​ .
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Answered by kate200468 | 2024-12-23