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Find cosθ, cotθ, and secθ, where θ is the angle shown in the figure. Give exact values, not decimal approximations.cosθ=cotθ=secθ=

Find Cosθ Cotθ And Secθ Where Θ Is The Angle Shown In The Figure Give Exact Values Not Decimal Approximationscosθcotθsecθ class=

Sagot :

First let's find the missing value of the hypotenuse:

[tex]\begin{gathered} c^2=a^2+b^2 \\ a=4 \\ b=5 \\ \Rightarrow c^2=(4)^2+(5)^2=16+25=41 \\ \Rightarrow c=\sqrt[]{41} \\ \end{gathered}[/tex]

we have that the hypotenuse equals sqrt(41). Now we can find the values of the trigonometric functions:

[tex]\begin{gathered} \cos (\theta)=\frac{adjacent\text{ side}}{hypotenuse} \\ \Rightarrow\cos (\theta)=\frac{4}{\sqrt[]{41}} \\ \sec (\theta)=\frac{1}{\cos (\theta)} \\ \Rightarrow\sec (\theta)=\frac{1}{\frac{4}{\sqrt[]{41}}}=\frac{\sqrt[]{41}}{4} \\ \tan (\theta)=\frac{opposite\text{ side}}{adjacent\text{ side}} \\ \Rightarrow\tan (\theta)=\frac{5}{4} \\ \cot (\theta)=\frac{1}{\tan (\theta)} \\ \Rightarrow\cot (\theta)=\frac{1}{\frac{5}{4}}=\frac{4}{5} \end{gathered}[/tex]

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