Question

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

Answer 1

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

`\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}`

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

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