Question

How do you find the remaining trigonometric functions of \[\theta \] given \[\cos \theta = - \dfrac{{20}}{{29}}\] and \[\theta \] terminates in QII?

Answer 1

Hint: In this question, we are given the value of \[\cos \theta \] and we are asked to find the all other trigonometric ratio. We have to find all the trigonometric ratios one by one. Using the identity \[{\sin ^2}\theta + {\cos ^2}\theta = 1\], we can find the \[\sin \theta \] can be obtained. Now we have the value of \[\sin \theta \] and \[\cos \theta \], by these values \[\tan \theta \] can be calculated, and other trigonometric ratios can be calculated by finding the reciprocal of these trigonometric ratios.

Complete step-by-step answer:
The angles which lie between ${0^o}$ and ${90^o}$ are said to lie in the first quadrant. The angles between \[{90^o}\] and \[{180^o}\] are in the second quadrant, angles between \[{180^o}\] and \[{270^o}\] are in the third quadrant and angles between \[{270^o}\] and \[{360^o}\] are in the fourth quadrant.
In the first quadrant, the values for sin, cos and tan are positive.
In the second quadrant, the values for sin are positive only.
In the third quadrant, the values for tan are positive only.
In the fourth quadrant, the values for cos are positive only.
Now given value is \[\cos \theta = - \dfrac{{20}}{{29}}\],
Now using the identity \[{\sin ^2}\theta + {\cos ^2}\theta = 1\],
Now substituting the value of \[\cos \theta \] in the identity, we get,
\[ \Rightarrow {\sin ^2}\theta + {\left( {\dfrac{{ - 20}}{{29}}} \right)^2} = 1\],
Now squaring we get,
\[ \Rightarrow {\sin ^2}\theta + \dfrac{{400}}{{841}} = 1\],
Now taking the constant value to the right hand side, we get,’
\[ \Rightarrow {\sin ^2}\theta = 1 - \dfrac{{400}}{{841}}\],
Now simplifying we get,
\[ \Rightarrow {\sin ^2}\theta = \dfrac{{841 - 400}}{{841}}\],
Again simplifying we get,
\[ \Rightarrow {\sin ^2}\theta = \dfrac{{441}}{{841}}\],
Now taking square root we get,
\[ \Rightarrow \sin \theta = \sqrt {\dfrac{{441}}{{841}}} \],
Simplifying we get,
\[ \Rightarrow \sin \theta = \dfrac{{21}}{{29}}\],
As the value of \[\theta \] ends at quadrant II so sin values are positive.
Now using the trigonometric identities we get,
\[ \Rightarrow \tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}\],
Now substituting the value of \[\sin \theta \] and \[\cos \theta \] we get,
\[ \Rightarrow \tan \theta = \dfrac{{\dfrac{{21}}{{29}}}}{{\dfrac{{ - 20}}{{29}}}}\],
Now simplifying we get,
\[ \Rightarrow \tan \theta = - \dfrac{{21}}{{20}}\],
As the value of \[\theta \] ends at quadrant II so tan values are negative.
Now as \[\cot \theta \] is the inverse of \[\tan \theta \], we get,
\[ \Rightarrow \cot \theta = \dfrac{1}{{\tan \theta }}\],
Now substituting the value of \[\tan \theta \], we get,
\[ \Rightarrow \cot \theta = \dfrac{1}{{ - \dfrac{{21}}{{20}}}}\],
Now simplifying we get,
\[ \Rightarrow \cot \theta = - \dfrac{{20}}{{21}}\],
As the value of \[\theta \] ends at quadrant II so cot values are negative.
Now we know that \[\csc \theta \] is the inverse of \[\sin \theta \], we get,
\[ \Rightarrow \csc \theta = \dfrac{1}{{\sin \theta }}\],
Now substituting the value of \[\sin \theta \], we get,
\[ \Rightarrow \csc \theta = \dfrac{1}{{\dfrac{{21}}{{29}}}}\],
Now simplifying we get,
\[ \Rightarrow \csc \theta = \dfrac{{29}}{{21}}\],
As the value of \[\theta \] ends at quadrant II so csc values are positive.
And we know that \[\sec \theta \] is the inverse of \[\cos \theta \] we get,
\[ \Rightarrow \sec \theta = \dfrac{1}{{\cos \theta }}\],
Now substituting the value of \[\cos \theta \] we get,
\[ \Rightarrow \sec \theta = \dfrac{1}{{ - \dfrac{{20}}{{29}}}}\],
Now simplifying we get,
\[ \Rightarrow \sec \theta = - \dfrac{{29}}{{20}}\].
As the value of \[\theta \] ends at quadrant II so sec values are negative.
The value of trigonometric ratios are, \[\cos \theta = - \dfrac{{20}}{{29}}\],\[\sin \theta = \dfrac{{21}}{{29}}\], \[\tan \theta = - \dfrac{{21}}{{20}}\], \[\cot \theta = - \dfrac{{20}}{{21}}\],\[\sec \theta = - \dfrac{{29}}{{20}}\],\[\csc \theta = \dfrac{{29}}{{21}}\].
Final Answer:

\[\therefore \] The value of trigonometric ratios are if \[\cos \theta = - \dfrac{{20}}{{29}}\], are\[\sin \theta = \dfrac{{21}}{{29}}\],\[\tan \theta = - \dfrac{{21}}{{20}}\],\[\cot \theta = - \dfrac{{20}}{{21}}\],\[\sec \theta = - \dfrac{{29}}{{20}}\],\[\csc \theta = \dfrac{{29}}{{21}}\].

Note:
Most of the trigonometry calculations are done by using trigonometric ratios. There are 6 trigonometric ratios present in trigonometry. Every other important trigonometry formula is derived with the help of these ratios.
The 6 important ratios named as sin, cos, tan, sec, cot, sec. Sin and cos are fundamental or basic ratios whereas Tan, sec, cot, and csc are derived functions.