Question

Assume that $\mathrm{\theta }$ is an acute angle in a right triangle satisfying the given condition. Evaluate the remaining trigonometric function ? $\sin \mathrm{\theta }=\frac{4}{11}$

$\cos \mathrm{\theta }$$=$

$\tan \mathrm{\theta }=$

$\csc \mathrm{\theta }=$

$\sec \mathrm{\theta }=$

$\cot \mathrm{\theta }=$

Answer 1

Evaluate the remaining trigonometric function:

It is given that $\sin \mathrm{\theta }=\frac{4}{11}$.

Step-1: Value of $\cos \mathrm{\theta }$:

Use the following identity to find $\cos \mathrm{\theta }$:

${\mathbf{sin}}^{\mathbf{2}}\mathbf{\theta }\mathbf{+}{\mathbf{cos}}^{\mathbf{2}}\mathbf{\theta }\mathbf{=}\mathbf{1}$

Substitute the value of $\sin \mathrm{\theta }$ in it:

$\begin{array}{rcl}{\left(\frac{4}{11}\right)}^2+{\cos }^2\mathrm{\theta }& =& 1\\ & \Rightarrow & {\cos }^2\mathrm{\theta }=1-\frac{16}{121}\end{array}$

$\Rightarrow \cos \mathrm{\theta }=\frac{\sqrt{105}}{11}$

Step-2: Value of $\tan \mathrm{\theta }$:

Now, use the following identity to find $\tan \mathrm{\theta }$:

$\mathbf{tan}\mathbf{}\mathbf{\theta }\mathbf{=}\frac{\mathbf{sin}\mathbf{}\mathbf{\theta }}{\mathbf{cos}\mathbf{}\mathbf{\theta }}$ (substitute the value of $\sin \mathrm{\theta }$ and $\cos \mathrm{\theta }$ in it)

$\begin{array}{rcl}\tan \theta & =& \frac{{\displaystyle \frac{4}{11}}}{{\displaystyle \frac{\sqrt{105}}{11}}}\\ & =& \frac{4}{\sqrt{105}}\\ & =& \frac{4}{\sqrt{105}}\times \frac{\sqrt{105}}{\sqrt{105}}\\ & =& \frac{4\sqrt{105}}{105}\end{array}$

Step-3: Value of $\text{cot θ,cosec θ,sec θ}$:

$cot\mathrm{\theta }=\frac{1}{{\displaystyle \frac{4}{\sqrt{105}}}}=\frac{\sqrt{105}}{4}$ $\left[\cot \theta =\frac{1}{\tan \theta }\right]$

$\csc \mathrm{\theta }=\frac{1}{{\displaystyle \frac{4}{11}}}=\frac{11}{4}$ $\left[csc\theta =\frac{1}{sin\theta }\right]$

$\begin{array}{rcl}sec\theta & =& \frac{1}{\cos \theta }\\ & =& \frac{1}{{\displaystyle \frac{\sqrt{105}}{11}}}\\ & =& \frac{11\sqrt{105}}{105}\end{array}$ $\left[\sec \theta =\frac{1}{\cos \theta }\right]$

Hence, the remaining trigonometric function are:

$$\begin{gathered} \mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathbf{\theta }\mathbf{=}\frac{\sqrt{\mathbf{105}}}{\mathbf{11}} \\ \mathbf{tan}\mathbf{}\mathbf{\theta }\mathbf{=}\frac{\mathbf{4}\sqrt{\mathbf{105}}}{\mathbf{105}} \\ \mathbf{csc}\mathbf{}\mathbf{\theta }\mathbf{=}\frac{\mathbf{11}}{\mathbf{4}}\mathbf{=}\mathbf{2}\frac{\mathbf{3}}{\mathbf{4}} \\ \mathbf{sec}\mathbf{}\mathbf{\theta }\mathbf{=}\frac{\mathbf{11}\sqrt{\mathbf{105}}}{\mathbf{105}} \\ \mathbf{cot}\mathbf{}\mathbf{\theta }\mathbf{=}\frac{\sqrt{\mathbf{105}}}{\mathbf{4}} \end{gathered}$$