Question

If \[5\sec \theta -12\cos ec\theta =0\] , then find the values of sec \[\theta \] , cos \[\theta \] and sin \[\theta \].

Answer 1

Hint: In such questions where sec \[\theta \] and cosec \[\theta \] are involved, we first try to simplify the equation or the expression using the following relations of trigonometric functions which are as follows
\[\sec \theta =\dfrac{1}{\cos \theta }\] and \[\cos ec\theta =\dfrac{1}{\sin \theta }\] .

Complete step-by-step answer:
As mentioned in the question, we must proceed as given in the hint as follows
\[\begin{align}
  & 5\sec \theta -12\cos ec\theta =0 \\
 & 5\dfrac{1}{\cos \theta }-12\dfrac{1}{\sin \theta }=0 \\
 & 5\sin \theta =12\cos \theta \\
 & \tan \theta =\dfrac{12}{5} \\
\end{align}\]

Now, as we have got tan \[\theta \] , we will use the basic definition of tan \[\theta \] to evaluate the other values that are asked in the question as follows
\[\begin{align}
  & \tan \theta =\dfrac{12}{5} \\
 & \tan \theta =\dfrac{perpendicular}{base} \\
\end{align}\]

Now, we know that perpendicular = 12 and base = 5, therefore, using the Pythagoras theorem to get the hypotenuse, we get
\[\begin{align}
  & {{\left( perpendicular \right)}^{2}}+{{\left( base \right)}^{2}}={{(hypotenuse)}^{2}} \\
 & {{\left( 12 \right)}^{2}}+{{\left( 5 \right)}^{2}}={{(hypotenuse)}^{2}} \\
 & {{(hypotenuse)}^{2}}=169 \\
\end{align}\]

Now, on taking square root of both the side, we get
Hypotenuse=13

Now,
\[\begin{align}
  & \cos \theta =\dfrac{base}{hypotenuse}=\dfrac{5}{13} \\
 & \sec \theta =\dfrac{1}{\cos \theta }=\dfrac{13}{5} \\
 & \sin \theta =\dfrac{perpendicular}{hypotenuse}=\dfrac{12}{13} \\
\end{align}\]

Note: In such questions where sec \[\theta \] and cosec \[\theta \] are involved, it is advisable that first we should first try to simplify the equation or the expression using the following relations of trigonometric functions which are as follows
\[\sec \theta =\dfrac{1}{\cos \theta }\] and \[\cos ec\theta =\dfrac{1}{\sin \theta }\] .
For question that involve tan \[\theta \] and cot \[\theta \] , we first try to simplify the equation or the expression using the following relations of trigonometric functions which are as follows
\[\tan \theta =\dfrac{\sin \theta }{\cos \theta }\] and \[\cot \theta =\dfrac{\cos \theta }{\sin \theta }\] .
By following this approach, there are less chances of committing a mistake by the students as this method brings the equations and expressions into the most basic forms and in this form solving the question becomes a lot easier.