Question

Find csc, cot0, and sin 0, where is the angle shown in the figure.Give exact values, not decimál approximations.

Answer 1

Step-by-step explanation:

Given that:

`\begin{gathered} \theta=? \\ Hypotenuse=6 \\ Adjacent=5 \end{gathered}`

We will obtain the following trigonometric identities as shown below:

`\begin{gathered} csc\theta=(1)/(\sin\theta) \\ \text{Using Pythagoras Theorem, we will obtain the missing side:} \\ a^2=c^2-b^2 \\ a^2=6^2-5^2 \\ a^2=36-25 \\ a^2=11 \\ a=√(11) \\ a=opposite=√(11) \\ \\ \sin\theta=(opposite)/(hypotenuse)=(√(11))/(6) \\ csc\theta=(1)/((√(11))/(6))=(6)/(√(11)) \\ csc\theta=(6)/(√(11)) \end{gathered}`

Then,

`\begin{gathered} cot\theta=(1)/(\tan\theta) \\ \tan\theta=(opposite)/(adjacent)=(√(11))/(5) \\ cot\theta=(1)/((√(11))/(5)) \\ cot\theta=(5)/(√(11)) \end{gathered}`

Then,

`\sin\theta=(√(11))/(6)`