Trigonometric functions of $\left( {{90}^{\circ }}+\theta \right)$ and $\left( 180-\theta \right)$ and $\left( 180+\theta \right)$.

Hint: Use the fundamental concepts of conversion of trigonometric functions by changing the angles. All the trigonometric functions are positive in the first quadrant, $\sin ,\operatorname{cs}c$ are positive in second, $tan,\cot $ are positive in third, $cos,sec$are positive in fourth quadrant. Angle in addition or subtraction with ${{90}^{\circ }}$, will convert $\sin \to \cos ,\cot \to \tan ,\csc \to \sec $ and vice-versa as well, but with ${{180}^{\circ }}$, function will not change. Sign while converting the function, will be decided by the quadrant in which angle is lying and if that given function is positive in that sign will be positive or negative.

Complete step-by-step answer:



We need to know some rules for the conversion of one trigonometric function to another. First of all, we need to know that, all the trigonometric functions are positive in first quadrant i.e. between ${{0}^{\circ }}\to {{90}^{\circ }}$, and in second quadrant, only sine and cosine function will give positive value and others will have negative value i.e. between ${{90}^{\circ }},{{180}^{\circ }}$. Similarly, in the third quadrant $\tan ,\cot $ functions are positive and others are negative i.e. between ${{180}^{\circ }}\to {{270}^{\circ }}$. And, hence, $\cos ,\sec $ will be positive in the fourth quadrant and others will be negative.
Now, we need to know some rules for conversion of trigonometric functions by changing their angles by adding or subtracting ${{90}^{\circ }},{{180}^{\circ }},{{360}^{\circ }},{{270}^{\circ }}$ or any other multiple of ${{90}^{\circ }}$.

So, rules are given as:

If any trigonometric function will have angle in addition or subtraction with ${{90}^{\circ }}$or multiple of ${{90}^{\circ }}$ (not multiple of ${{180}^{\circ }}$), then we can do conversion by replacing

$\begin{align}

  & \sin \to \cos \\

 & \sec \to \csc \\

 & \tan \to \cot \\

\end{align}$

Or vice-versa of above will also be true.
And the sign of the converted value will be determined by the sign of the function going to be replaced in the quadrant with the help of the angle provided with it.

Example: $\sin \left( 90-\theta \right)=\cos \theta $

Hence, $\theta $ is subtracted from ${{90}^{\circ }}$

So, we need to replace $\sin \to \cos $ and as $90-\theta $ is lying in the first quadrant, and the $\sin $ function will be positive in that quadrant.

So, just replace $\sin \left( 90-\theta \right)\to \cos \theta $.

If any trigonometric function will have angle in addition or subtraction with ${{180}^{\circ }}$ or multiple of ${{180}^{\circ }}$, then we do not need to replace the given trigonometric function by any other function. And sign will be determined by the same approach given in rule 1.

So, let us write the trigonometric functions of $\left( 90+\theta \right),\left( 180-\theta \right),\left( 180+\theta \right)$ as:

$\begin{align}

  & \sin \left( 90+\theta \right)=\cos \theta ,\cos \left( 90+\theta \right)=-\sin \theta \\

 & tan\left( 90+\theta \right)=-\cot \theta ,\cot \left( 90+\theta \right)=-\tan \theta \\

 & \sec \left( 90+\theta \right)=-\csc \theta ,\csc \left( 90+\theta \right)=\sec \theta \\

\end{align}$

$\begin{align}

  & \sin \left( 180-\theta \right)=sin\theta ,\cos \left( 180-\theta
\right)=-co\operatorname{s}\theta \\

 & tan\left( 180-\theta \right)=-\tan \theta ,\cot \left( 180-\theta \right)=-\cot \theta \\

 & c\operatorname{sc}\left( 180-\theta \right)=csc\theta ,sec\left( 180-\theta \right)=-\sec
\theta \\

 & \sin \left( 180+\theta \right)=-sin\theta ,\cos \left( 180+\theta
\right)=-co\operatorname{s}\theta \\

 & tan\left( 180+\theta \right)=tan\theta ,\cot \left( 180+\theta \right)=-\cot \theta \\

 & c\operatorname{sc}\left( 180+\theta \right)=-csc\theta ,sec\left( 180+\theta
\right)=-\sec \theta \\

\end{align}$

Note: One may have question that, the multiples of 180 is also a multiple of ${{90}^{\circ }}$, so, whether we can apply trigonometric relations of addition o subtraction with ${{90}^{\circ }}$, or not. So, answer is yes, but try to split the angle of any trigonometric function in multiple of ${{180}^{\circ }}$, if angle is higher than ${{180}^{\circ }}$.
Don’t memorize the formulae, learn the rules given in the problem.