Question

Find out the values of angles $120^\circ$, $- 135^\circ$, $150^\circ$, $180^\circ$, $270^\circ$ for all the six trigonometric ratios.

Answer 1

Here, in the given question, we need to find the values of angles $120^\circ$, $- 135^\circ$, $150^\circ$, $180^\circ$, $270^\circ$ for all the six trigonometric ratios. We will use trigonometric formulas to get our required answer.
Formulae used:
$\sin \left( {90^\circ + \theta } \right) = \cos \theta$
$\cos \left( {90^\circ + \theta } \right) = - \sin \theta$
$\tan \left( {90^\circ + \theta } \right) = - \cot \theta$
$\cos ec\left( {90^\circ + \theta } \right) = \sec \theta$
$\sec \left( {90^\circ + \theta } \right) = - \cos ec\theta$
$\cot \left( {90^\circ + \theta } \right) = - \tan \theta$
$\sin \left( {90^\circ - \theta } \right) = \cos \theta$
$\sin \left( {90^\circ - \theta } \right) = \cos \theta$
$\cos \left( {90^\circ - \theta } \right) = \sin \theta$
$\tan \left( {90^\circ - \theta } \right) = \cot \theta$
$\cos ec\left( {90^\circ - \theta } \right) = \sec \theta$
$\sec \left( {90^\circ - \theta } \right) = \cos ec\theta$
$\cot \left( {90^\circ - \theta } \right) = \tan \theta$

Complete step by step answer:
$120^\circ$
Given below is the value of $120^\circ$ for all the trigonometric ratios.
$\Rightarrow \sin 120^\circ = \sin \left( {90^\circ + 30^\circ } \right) = \cos 30^\circ = \dfrac{{\sqrt 3 }}{2}$ (Value of $\cos 30^\circ = \dfrac{{\sqrt 3 }}{2}$ )
$\Rightarrow \cos 120^\circ = \cos \left( {90^\circ + 30^\circ } \right) = - \sin 30^\circ = - \dfrac{1}{2}$ (Value of $\sin 30^\circ = \dfrac{1}{2}$ )
$\Rightarrow \tan 120^\circ = \tan \left( {90^\circ + 30^\circ } \right) = - \cot 30^\circ = - \sqrt 3$ (Value of $\cot 30^\circ = \sqrt 3$ )
$\Rightarrow \cos ec120^\circ = \cos ec\left( {90^\circ + 30^\circ } \right) = \sec 30^\circ = \dfrac{2}{{\sqrt 3 }}$ (Value of $\sec 30^\circ = \dfrac{2}{{\sqrt 3 }}$ )
$\Rightarrow \sec 120^\circ = \sec \left( {90^\circ + 30^\circ } \right) = - \cos ec30^\circ = - 2$ (Value of $\cos ec30^\circ = 2$ )
$\Rightarrow \cot 120^\circ = \cot \left( {90^\circ + 30^\circ } \right) = - \tan 30^\circ = - \dfrac{1}{{\sqrt 3 }}$ (Value of $\tan 30^\circ = \dfrac{1}{{\sqrt 3 }}$ )
$- 135^\circ$
Given below is the value of $- 135^\circ$ for all the trigonometric ratios.
$\Rightarrow \sin \left( { - 135^\circ } \right) = - \sin 135^\circ = - \sin \left( {1 \times 90^\circ + 45^\circ } \right) = - \cos 45^\circ = - \dfrac{1}{{\sqrt 2 }}$ (Value of $\cos 45^\circ = \dfrac{1}{{\sqrt 2 }}$ )
$\Rightarrow \cos \left( { - 135^\circ } \right) = \cos 135^\circ = \cos \left( {1 \times 90^\circ + 45^\circ } \right) = - \sin 45^\circ = - \dfrac{1}{{\sqrt 2 }}$ (Value of $\sin 45^\circ = \dfrac{1}{{\sqrt 2 }}$ )
$\Rightarrow \tan \left( { - 135^\circ } \right) = - \tan 135^\circ = - \tan \left( {1 \times 90^\circ + 45^\circ } \right) = - \left( { - \cot 45} \right)^\circ = 1$ (Value of $\cot 45^\circ = 1$ )
$\Rightarrow \cos ec\left( { - 135^\circ } \right) = - \cos ec135^\circ = - \cos ec\left( {1 \times 90^\circ + 45^\circ } \right) = - sec45^\circ = - \sqrt 2$ (Value of $\sec 45^\circ = \sqrt 2$ )
$\Rightarrow \sec \left( { - 135^\circ } \right) = \sec 135^\circ = \sec \left( {1 \times 90^\circ + 45^\circ } \right) = - \cos ec45^\circ = - \sqrt 2$ (Value of $\cos ec45^\circ = - \sqrt 2$ )
$\Rightarrow \cot \left( { - 135^\circ } \right) = - \cot 135^\circ = - \cot \left( {1 \times 90^\circ + 45^\circ } \right) = - \left( { - \tan 45^\circ } \right) = 1$ (Value of $\tan 45^\circ = 1$ )
$150^\circ$
Given below is the value of $150^\circ$ for all the trigonometric ratios.
$\Rightarrow \sin 150^\circ = \sin \left( {2 \times 90^\circ - 30^\circ } \right) = \sin 30^\circ = \dfrac{1}{2}$ (Value of $\sin 30^\circ = \dfrac{1}{2}$ )
$\Rightarrow \cos 150^\circ = \cos \left( {2 \times 90^\circ - 30^\circ } \right) = \cos 30^\circ = - \dfrac{{\sqrt 3 }}{2}$ (Value of $\cos 30^\circ = \dfrac{{\sqrt 3 }}{2}$ )
$\Rightarrow \tan 150^\circ = \tan \left( {2 \times 90^\circ - 30^\circ } \right) = - \tan 30^\circ = - \dfrac{1}{{\sqrt 3 }}$ (Value of $\tan 30^\circ = \dfrac{1}{{\sqrt 3 }}$ )
$\Rightarrow \cos ec150^\circ = \cos ec\left( {2 \times 90^\circ - 30^\circ } \right) = \cos ec30^\circ = 2$ (Value of $\cos ec30^\circ = 2$ )
$\Rightarrow \sec 150^\circ = \sec \left( {2 \times 90^\circ - 30^\circ } \right) = \sec 30^\circ = \dfrac{2}{{\sqrt 3 }}$ (Value of $\sec 30^\circ = \dfrac{2}{{\sqrt 3 }}$ )
$\Rightarrow \cot 150^\circ = \cot \left( {2 \times 90^\circ - 30^\circ } \right) = - \cot 30^\circ = - \sqrt 3$ (Value of $\cot 30^\circ = \sqrt 3$ )
$180^\circ$
Given below is the value of $180^\circ$ for all the trigonometric ratios.
$\Rightarrow \sin 180^\circ = \sin \left( {2 \times 90^\circ - 0^\circ } \right) = \sin 0^\circ = 0$ (Value of $\sin 0^\circ = 0$ )
$\Rightarrow \cos 180^\circ = \cos \left( {2 \times 90^\circ - 0^\circ } \right) = - \cos 0^\circ = - 1$ (Value of $\cos 0^\circ = 1$ )
$\Rightarrow \tan 180^\circ = \tan \left( {2 \times 90^\circ + 0^\circ } \right) = \tan 0^\circ = 0$ (Value of $\tan 0^\circ = 0$ )
$\Rightarrow \cos ec180^\circ = \cos ec\left( {2 \times 90^\circ - 0^\circ } \right) = \cos ec0^\circ = Undefined$ (Value of $\cos ec0^\circ = Undefined$ )
$\Rightarrow \sec 180^\circ = \sec \left( {2 \times 90^\circ - 0^\circ } \right) = - \sec 0^\circ = - 1$ (Value of $\sec 0^\circ = 1$ )
$\Rightarrow \cot 180^\circ = \cot \left( {2 \times 90^\circ + 0^\circ } \right) = \cot 0^\circ = Undefined$ (Value of $\cot 0^\circ = Undefined$ )
$270^\circ$
Given below is the value of $270^\circ$ for all the trigonometric ratios.
$\Rightarrow \sin 270^\circ = \sin \left( {3 \times 90^\circ + 0^\circ } \right) = - \cos 0^\circ = - 1$ (Value of $\cos 0^\circ = 1$ )
$\Rightarrow \cos 270^\circ = \cos \left( {3 \times 90^\circ + 0^\circ } \right) = \sin 0^\circ = 0$ (Value of $\sin 0^\circ = 0$ )
$\Rightarrow \tan 270^\circ = \tan \left( {3 \times 90^\circ + 0^\circ } \right) = - \cot 0^\circ = Undefined$ (Value of $\cot 0^\circ = Undefined$ )
$\Rightarrow \cos ec270^\circ = \cos ec\left( {3 \times 90^\circ + 0^\circ } \right) = - \sec 0^\circ = - 1$ (Value of $\sec 0^\circ = 1$ )
$\Rightarrow \sec 270^\circ = \sec \left( {3 \times 90^\circ + 0^\circ } \right) = \cos ec0^\circ = Undefined$ (Value of $\cos ec0^\circ = Undefined$ )
$\Rightarrow \cot 270^\circ = \cot \left( {3 \times 90^\circ + 0^\circ } \right) = - \tan 0^\circ = 0$ (Value of $\tan 0^\circ = 0$ )

Note:
To solve this type of questions, one must remember all the formulae. We can also find the trigonometric values using the $\sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B$ formula, by this formula we can find the trigonometric value of $\sin e$. After this, we can use the formula ${\cos ^2}x + {\sin ^2}x = 1$ to find the value of $\cos$. We know that $\tan x = \dfrac{{\sin x}}{{\cos x}}$ hence we can find the value of $\tan$. We also know that $\cot$, $\sec$ and $\cos ec$ are reciprocal of $\tan$, $\cos$ and $\sin e$ respectively. Hence we can easily find the value of $\cot$, $\sec$ and $\cos ec$. Hence we will get all the required values.