Allied Angles (Part 1)

Two angles are called allied if their sum or difference is a multiple of π/2   

• sin (90⁰ - θ) = cos (θ)

• cos (90⁰ - θ) = sin (θ)

• tan (90⁰ - θ) = cot (θ)

• csc (90⁰ - θ) = sec (θ)          

• sec (90⁰ - θ) = csc (θ)

• cot (90⁰ - θ) = tan (θ)

• sin (90⁰ + θ) = cos (θ)

• cos (90⁰ + θ) = - sin (θ)

• tan (90⁰ + θ) = - cot (θ)

• csc (90⁰ + θ) = sec (θ)          

• sec (90⁰ + θ) = - csc (θ)

• cot (90⁰ + θ) = - tan (θ)

Allied Angles (Part 2)

$\begin{aligned} & \sin \left(180^{\circ}-\theta\right)=\sin (\theta) \\ & \cos \left(180^{\circ}-\theta\right)=-\cos (\theta) \\ & \tan \left(180^{\circ}-\theta\right)=-\tan (\theta) \\ & \csc \left(180^{\circ}-\theta\right)=\csc (\theta) \\ & \sec \left(180^{\circ}-\theta\right)=-\sec (\theta) \\ & \cot \left(180^{\circ}-\theta\right)=-\cot (\theta) \\ & \sin \left(180^{\circ}+\theta\right)=-\sin (\theta) \\ & \cos \left(180^{\circ}+\theta\right)=-\cos (\theta) \\ & \tan \left(180^{\circ}+\theta\right)=\tan (\theta) \\ & \csc \left(180^{\circ}+\theta\right)=-\csc (\theta) \\ & \sec \left(180^{\circ}+\theta\right)=-\sec (\theta) \\ & \cot \left(180^{\circ}+\theta\right)=\cot (\theta)\end{aligned}$

AID TO REMEMBER

All the trigonometric functions of a real number of form $2 n(\pi / 2) \pm x(n \in I)$ (i.e. an even multiple of $\pi / 2 \pm x$ ) are numerically equal to the same function of $x$, with sign depending on the quadrant in which terminal side of the angles lies.

For example $\cos (\pi+x)=\cos (2(\pi / 2)+x)=-\cos (x)$, -ve sign was chosen because $(\pi+x)$ lies in 3rd quadrant and 'cos' is -ve in the third quadrant.

All the trigonometric functions of a real number of the form $(2 n+1) \pi / 2 \pm x(n \in I)$ (i.e. an even multiple of $\pi / 2 \pm x)$ is numerically equal to the co-function of $x$, with sign depending on the quadrant in which terminal side of the angles lies.

Note that 'sin' and 'cos' are co-functions of each other, 'tan' and 'cot' are co-functions of each other and 'sec' and 'cosec' are cofunctions of each other.

For example: $\sec (\pi / 2+x)=-\operatorname{cosec}(x)$, as $(\pi / 2+x)$ lies in the 2 nd quadrant and 'sec' is -ve in $2 n d$ quadrant.