Question

If \[\tan \theta =\dfrac{15}{8}\], find the value of all other T-ratios of \[\theta \] .

Answer 1

Hint: In this question, we are given the value of \[\tan \theta \] , hence, we can use the following relations which involve \[\tan \theta \] and hence, we can get the values of all other T-ratios.

Complete step-by-step answer:
\[\begin{align}
  & {{\sec }^{2}}\theta -{{\tan }^{2}}\theta =1 \\
 & \cos \theta =\dfrac{1}{\sec \theta } \\
 & \sin \theta =\tan \theta \cdot \cos \theta \\
 & \cos ec\theta =\dfrac{1}{\sin \theta } \\
 & \cot \theta =\dfrac{1}{\tan \theta } \\
\end{align}\]
Now, by using these above relations, we can get the required results and values.
Now, from the given question we have
\[\tan \theta =\dfrac{15}{8}\ \ \ \ \ ...(a)\]
Now, by using the trigonometric identity which gives the relation between the function that are mentioned in the hint, we get the following substitute the value from the question and as well as from equation (a)
\[\begin{align}
  & \Rightarrow {{\sec }^{2}}\theta -{{\tan }^{2}}\theta =1 \\
 & \Rightarrow {{\sec }^{2}}\theta ={{\tan }^{2}}\theta +1 \\
 & \Rightarrow {{\sec }^{2}}\theta ={{\left( \dfrac{15}{8} \right)}^{2}}+1 \\
 & \Rightarrow {{\sec }^{2}}\theta =\left( \dfrac{225}{64}+1 \right) \\
 & \Rightarrow {{\sec }^{2}}\theta =\left( \dfrac{225+64}{64} \right) \\
 & \Rightarrow {{\sec }^{2}}\theta =\left( \dfrac{289}{64} \right) \\
 & \Rightarrow \sec \theta =\sqrt{\dfrac{289}{64}}=\dfrac{17}{8} \\
\end{align}\]
For finding the value of cos function, we could again use the relations given in the hint as follows
\[\begin{align}
  & \Rightarrow \left( \dfrac{1}{\sec \theta } \right)=\cos \theta \\
 & \Rightarrow \cos \theta =\left( \dfrac{1}{\dfrac{17}{8}} \right)=\dfrac{8}{17} \\
\end{align}\]
Now, again from the hint, we know that we can get the sin function as follows
\[\begin{align}
  & \Rightarrow \sin \theta =\tan \theta \cdot \cos \theta \\
 & \Rightarrow \sin \theta =\dfrac{15}{8}\times \dfrac{8}{17} \\
 & \Rightarrow \sin \theta =\dfrac{15}{17} \\
\end{align}\]
Now, again referring to the relations, we get the value of cosec function as follows
\[\begin{align}
  & \Rightarrow \cos ec\theta =\dfrac{1}{\sin \theta } \\
 & \Rightarrow \cos ec\theta =\dfrac{1}{\dfrac{15}{17}} \\
 & \Rightarrow \cos ec\theta =\dfrac{17}{15} \\
\end{align}\]
Now, again referring to the relations, we get the value of cot function as follows
\[\begin{align}
  & \Rightarrow \tan \theta =\dfrac{1}{\cot \theta } \\
 & \Rightarrow \cot \theta =\dfrac{1}{\tan \theta } \\
 & \Rightarrow \cot \theta =\dfrac{1}{\dfrac{15}{8}} \\
 & \Rightarrow \cot \theta =\dfrac{8}{15} \\
\end{align}\]
Hence, these are the values of all the T-ratios of \[\theta \] using the value of the sin function that was given in the hint.

Note: It is very important to know all the relations between the different trigonometric functions because without knowing these relations, one could never get to the correct answer.
It is important to note that while calculating the values of respective functions we need to use the identities accordingly and solve them. Because neglecting any of the terms or writing it incorrectly changes the complete result.