Trigonometric Ratios in Radians

Introduction:

When you delve deeper into your study of trigonometry, you will find that when talking about angle measures, the radian scale is much more widespread than the degree scale. Given this, it is imperative to know the important T-ratio values in the radian scale. These are summarized below (both degree and radian scales are used; when the angle value has no unit specified, it is assumed to be in radians; N.D. means not-defined).
Make sure you read and commit to memory each of these values of trigonometric ratios.

Specific Degree measure in Radians:

1. \(\begin{align}\theta  = {0^0}\,\,\,{\rm{or}}\,\,\,\theta  = 0\end{align}\)

• \(\begin{align}\sin {0^0} = 0\,\,\,{\rm{or}}\,\,\,\sin 0 = 0\end{align}\)

• \(\begin{align}\cos {0^0} = 1\,\,\,{\rm{or}}\,\,\,\cos 0 = 1\end{align}\)

• \(\begin{align}\tan {0^0} = 0\,\,\,{\rm{or}}\,\,\,\tan 0 = 0\end{align}\)

• \(\begin{align}{\rm{cosec}}\,{0^0}:{\rm{N.D.}}\,\,\,{\rm{or}}\,\,\,{\rm{cosec}}\,0:{\rm{N.D.}}\end{align}\)

• \(\begin{align}\sec {0^0} = 1\,\,\,{\rm{or}}\,\,\,\sec 0 = 1\end{align}\)

• \(\begin{align}\cot {0^0}:{\rm{N.D.}}\,\,\,{\rm{or}}\,\,\,\cot 0:{\rm{N.D.}}\end{align}\)

 2. \(\begin{align}\theta  = {30^0}\,\,\,{\rm{or}}\,\,\,\theta  = \frac{\pi }{6}\end{align}\)

• \(\begin{align}\sin {30^0} = \frac{1}{2}\,\,\,{\rm{or}}\,\,\,\sin \frac{\pi }{6} = \frac{1}{2}\end{align}\)

• \(\begin{align}\cos {30^0} = \frac{{\sqrt 3 }}{2}\,\,\,{\rm{or}}\,\,\,\cos \frac{\pi }{6} = \frac{{\sqrt 3 }}{2}\end{align}\)

• \(\begin{align}\tan {30^0} = \frac{1}{{\sqrt 3 }}\,\,\,{\rm{or}}\,\,\,\tan \frac{\pi }{6} = \frac{1}{{\sqrt 3 }}\end{align}\)

• \(\begin{align} {\rm{cosec}}\,{30^0}{\rm{ = 2}}\,\,\,{\rm{or}}\,\,\, {\rm{cosec}}\frac{\pi }{6} = 2\end{align}\)

• \(\begin{align}\sec {30^0} = \frac{2}{{\sqrt 3 }}\,\,\,{\rm{or}}\,\,\,\sec \frac{\pi }{6} = \frac{2}{{\sqrt 3 }}\end{align}\)

• \(\begin{align}\cot {30^0} = \sqrt 3 \,\,\,{\rm{or}}\,\,\,\cot \frac{\pi }{6} = \sqrt 3 \end{align}\)

3. \(\begin{align}\theta  = {45^0}\,\,\,{\rm{or}}\,\,\,\theta  = \frac{\pi }{4}\end{align}\)

• \(\begin{align}\sin {45^0} = \frac{1}{{\sqrt 2 }}\,\,\,{\rm{or}}\,\,\,\sin \frac{\pi }{4} = \frac{1}{{\sqrt 2 }}\end{align}\)

• \(\begin{align}\cos {45^0} = \frac{1}{{\sqrt 2 }}\,\,\,{\rm{or}}\,\,\,\cos \frac{\pi }{4} = \frac{1}{{\sqrt 2 }}\end{align}\)

• \(\begin{align}\tan {45^0} = 1\,\,\,{\rm{or}}\,\,\,\tan \frac{\pi }{4} = 1\end{align}\)

• \(\begin{align} {\rm{cosec}}\,{45^0}{\rm{ = }}\sqrt {\rm{2}} \,\,\,{\rm{or}}\,\,\, {\rm{cosec}}\frac{\pi }{4} = \sqrt {\rm{2}} \end{align}\)

• \(\begin{align}\sec {45^0} = \sqrt {\rm{2}} \,\,\,{\rm{or}}\,\,\,\sec \frac{\pi }{4} = \sqrt {\rm{2}} \end{align}\)

• \(\begin{align}\cot {45^0} = 1\,\,\,{\rm{or}}\,\,\,\cot \frac{\pi }{4} = 1\end{align}\)

4.  \(\begin{align}\theta  = {60^0}\,\,\,{\rm{or}}\,\,\,\theta  = \frac{\pi }{3}\end{align}\)

• \(\begin{align}\sin {60^0} = \frac{{\sqrt 3 }}{2}\,\,\,{\rm{or}}\,\,\,\sin \frac{\pi }{3} = \frac{{\sqrt 3 }}{2}\end{align}\)

• \(\begin{align}\cos {60^0} = \frac{1}{2}\,\,\,{\rm{or}}\,\,\,\cos \frac{\pi }{3} = \frac{1}{2}\end{align}\)

• \(\begin{align}\tan {60^0} = \sqrt 3 \,\,\,{\rm{or}}\,\,\,\tan \frac{\pi }{3} = \sqrt 3 \end{align}\)

• \(\begin{align} {\rm{cosec}}\,{60^0}{\rm{ = }}\frac{2}{{\sqrt 3 }}\,\,\,{\rm{or}}\,\,\, {\rm{cosec}}\frac{\pi }{3} = \frac{2}{{\sqrt 3 }}\end{align}\)

• \(\begin{align}\sec {60^0} = {\rm{2}}\,\,\,{\rm{or}}\,\,\,\sec \frac{\pi }{3} = {\rm{2}}\end{align}\)

• \(\begin{align}\cot {60^0} = \frac{1}{{\sqrt 3 }}\,\,\,{\rm{or}}\,\,\,\cot \frac{\pi }{3} = \frac{1}{{\sqrt 3 }}\end{align}\)

5.  \(\begin{align}\theta  = {90^0}\,\,\,{\rm{or}}\,\,\,\theta  = \frac{\pi }{2}\end{align}\)

• \(\begin{align}\sin {90^0} = 1\,\,\,{\rm{or}}\,\,\,\sin \frac{\pi }{2} = 1\end{align}\)

• \(\begin{align}\cos {90^0} = 0\,\,\,{\rm{or}}\,\,\,\cos \frac{\pi }{2} = 0\end{align}\)

• \(\begin{align}\tan {90^0}:{\rm{nd}}\,\,\,{\rm{or}}\,\,\,\tan \frac{\pi }{2}:{\rm{nd}}\end{align}\)

• \(\begin{align}{\rm{cosec}}\,{90^0}{\rm{ = }}1\,\,\,{\rm{or}}\,\,\, {\rm{cosec}}\frac{\pi }{2} = 1\end{align}\)

• \(\begin{align}\sec {90^0}:{\rm{nd}}\,\,\,{\rm{or}}\,\,\,\sec \frac{\pi }{2}:{\rm{nd}}\end{align}\)

• \(\begin{align}\cot {90^0} = 0\,\,\,{\rm{or}}\,\,\,\cot \frac{\pi }{2} = 0\end{align}\)

Solved Example:

Example 1: Write the values of: 

(a) \(\begin{align}\sin \frac{\pi }{6}\end{align}\)

(b) \(\begin{align}\sec \frac{\pi }{3}\end{align}\)

(c) \(\begin{align}\cot \frac{\pi }{2}\end{align}\)

Solution: We have:

(a) \(\begin{align}\sin \frac{\pi }{6} = \sin {30^0} = \frac{1}{2}\end{align}\)

(b) \(\begin{align}\sec \frac{\pi }{3} = \sec {60^0} = 2\end{align}\)

(c) \(\begin{align}\cot \frac{\pi }{2} = \cot {90^0} = 0\end{align}\)

Challenge: Write the values of:

(1) \(\begin{align} {\rm{cosec}}\frac{\pi }{4}\end{align}\)

(2) \(\begin{align}\cos \frac{\pi }{2}\end{align}\)

(3) \(\begin{align}\tan \frac{\pi }{3}\end{align}\)

⚡Tip: Use a similar approach as in above example and also please go through trigonometric ratios of Specific Angles.