Question

If sin Θ = 5/6, what are the values of cos Θ and tan Θ

Answer 1

Given:

$\sin\theta=\frac{5}{6}$

Required- the values of cos θ and tan θ.

Explanation:

We know that in a right-angled triangle, for any angle θ, the value of sin θ, cos θ, and tan θ is:

$\begin{gathered} \sin\theta=\frac{Opposite}{Hypotenuse} \\ \\ \cos\theta=\frac{Adjacent}{Hypotenuse} \\ \\ \tan\theta=\frac{Opposite\text{ }}{Adjacent} \end{gathered}$

From the given value of sin θ, we get:

$\begin{gathered} Opposite\text{ =5} \\ \\ Hypotenuse\text{ =6} \end{gathered}$

Now, we calculate the value of Adjacent by the Pythagoras theorem as:

$\begin{gathered} (Hypotenuse)^2=(Adjacent)^2+(Opposite)^2 \\ \\ (6)^2=(Adjacent)^2+(5)^2 \\ \\ (Adjacent\text{ \rparen}^2\text{=36-25} \\ \\ Adjacent=\sqrt{11} \end{gathered}$

Now, the value of cos θ is:

$\begin{gathered} \cos\theta=\frac{Adjacent}{Hypotenuse} \\ \\ =\frac{\sqrt{11}}{6} \end{gathered}$

Now, the value of tan θ is:

$\begin{gathered} \tan\theta=\frac{Opposite}{Adjacent} \\ \\ =\frac{5}{\sqrt{11}} \\ \end{gathered}$

Final answer: The value of cos θ and tan θ is √11/6 and 5/√11 respectively.