Question

Find out the values of angle $270°$ 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 $270°$ is given.

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

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

So,

$$\begin{gathered} \sin (270°)=\sin (180°+90°) \\ \Rightarrow \sin (270°)=-\sin (90°) \\ \Rightarrow \sin (270°)=-1 \end{gathered}$$

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

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

So,

$$\begin{gathered} \cos (270°)=\cos (180°+90°) \\ \Rightarrow \cos (270°)=-\cos (90°) \\ \Rightarrow \cos (270°)=0 \end{gathered}$$

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,

$$\begin{gathered} \tan (270°)=\tan (180°+90°) \\ \Rightarrow \tan (270°)=\tan (90°) \\ \Rightarrow \tan (270°)=\text{N.D}\left\{Since,\tan (90°)=\text{N.D}\right\} \end{gathered}$$

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

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

So,

$$\begin{gathered} cot(270°)=cot(180°+90°) \\ \Rightarrow cot(270°)=cot(90°) \\ \Rightarrow cot(270°)=0 \end{gathered}$$

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,

$$\begin{gathered} sec(270°)=sec(180°+90°) \\ \Rightarrow sec(270°)=-sec(90°) \\ \Rightarrow sec(270°)=\text{N.D}\left\{\sin ce,sec(90°)=\text{N.D}\right\} \end{gathered}$$

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

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

So,

$$\begin{gathered} \cos ec(270°)=cosec(180°+90°) \\ \Rightarrow \cos ec(270°)=-cosec(90°) \\ \Rightarrow \cos ec(270°)=-1 \end{gathered}$$

Hence, the values of trigonometric ratios are

$$\begin{gathered} \sin {270}^{\circ }=-1 \\ \cos {270}^{\circ }=0 \\ \tan {270}^{\circ }=Undefined \\ \cos ec{270}^{\circ }=-1 \\ sec{270}^{\circ }=Undefined \\ cot{270}^{\circ }=0 \end{gathered}$$