Question

Find out the values of angle$180°$ for all the six trigonometric ratios.

Answer 1

Step 1: Compute the $\text{sine}$ and $\mathrm{cosine}$ of the given angle.

In the question, the measure of an angle $180°$ is given.

Compute the $\text{sine}$ of the given angle.

We know that, $\sin \left(180°-\theta \right)=\sin \left(\theta \right)$.

So,

$$\begin{gathered} \sin (180°)=\sin (180°-0°) \\ \Rightarrow \sin (180°)=\sin (0°) \\ \Rightarrow \sin (180°)=0 \end{gathered}$$

Compute the $\text{cosine}$ of the given angle.

We know that, $\cos \left(180°-\theta \right)=-\cos \left(\theta \right)$.

So,

$$\begin{gathered} \cos (180°)=\cos (180°-0°) \\ \Rightarrow \cos (180°)=-\cos (0°) \\ \Rightarrow \cos (180°)=-1 \end{gathered}$$

Step 2: Compute the $\text{tangent}$ and $\mathrm{cotangent}$ of the given angle.

Compute the $\text{tangent}$ of the given angle.

We know that, $\tan \left(180°-\theta \right)=-\tan \left(\theta \right)$.

So,

$$\begin{gathered} \tan (180°)=\tan (180°-0°) \\ \Rightarrow \tan (180°)=-\tan (0°) \\ \Rightarrow \tan (180°)=0 \end{gathered}$$

Compute the $\text{cotangent}$ of the given angle.

We know that, $cot\left(180°-\theta \right)=-cot\left(\theta \right)$.

So,

$$\begin{gathered} cot(180°)=cot(180°-0°) \\ \Rightarrow cot(180°)=-cot(0°) \\ \Rightarrow cot(180°)=\text{Not Define}\left\{cot\left(0°\right)=\text{Not Define}\right\} \end{gathered}$$

Step 3: Compute the $\text{secant}$ and $\text{cosecant}$ of the given angle.

Compute the $\text{secant}$ of the given angle.

We know that, $sec\left(180°-\theta \right)=-sec\left(\theta \right)$.

So,

$$\begin{gathered} sec(180°)=sec(180°-0°) \\ \Rightarrow sec(180°)=-sec(0°) \\ \Rightarrow sec(180°)=-1 \end{gathered}$$

Compute the $\text{cosecant}$ of the given angle.

We know that, $\cos ec\left(180°-\theta \right)=\cos ec\left(\theta \right)$.

So,

$$\begin{gathered} \cos ec(180°)=cosec(180°-0°) \\ \Rightarrow \cos ec(180°)=cosec(0°) \\ \Rightarrow \cos ec(180°)=\text{N.D}\left\{\cos ec(0°)=\text{N.D}\right\} \end{gathered}$$

Hence, the values of trigonometric ratios are

$$\begin{gathered} \sin {180}^{\circ }=0 \\ \cos {180}^{\circ }=-1 \\ \tan {180}^{\circ }=0 \\ c\mathrm{os}ec{180}^{\circ }=Undefined \\ sec{180}^{\circ }=-1 \\ cot{180}^{\circ }=Undefined \end{gathered}$$