Question

Find

(a).$\mathit{s}\mathit{i}\mathit{n}\mathit{\alpha }\mathit{a}\mathit{n}\mathit{d}\mathit{c}\mathit{o}\mathit{s}\mathit{\beta }$

(b).$\mathit{t}\mathit{a}\mathit{n}\mathit{\alpha }\mathit{a}\mathit{n}\mathit{d}\mathit{c}\mathit{o}\mathit{t}\mathit{\beta }$, and

(c). $\mathit{s}\mathit{e}\mathit{c}\mathit{\alpha }\mathit{a}\mathit{n}\mathit{d}\mathit{c}\mathit{s}\mathit{c}\mathit{\beta }\mathbf{.}$

Short answer

(a).

$\begin{array}{l}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\alpha }\mathbf{=}\frac{\mathbf{4}}{\mathbf{7}}\mathbf{\text{and}}\\ \mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\beta }\mathbf{=}\frac{\mathbf{4}}{\mathbf{7}}\mathbf{,}\mathbf{\text{forthetriangle}}\end{array}$

(b).

$\begin{array}{l}\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{\alpha }\mathbf{=}\frac{\mathbf{4}}{\sqrt{\mathbf{33}}}\mathbf{\text{and}}\\ \mathbf{c}\mathbf{o}\mathbf{t}\mathbf{\beta }\mathbf{=}\frac{\mathbf{4}}{\sqrt{\mathbf{33}}}\mathbf{,}\mathbf{\text{forthetriangle}}\end{array}$

(c).

$\begin{array}{l}\mathbf{s}\mathbf{e}\mathbf{c}\mathbf{\alpha }\mathbf{=}\frac{\mathbf{7}}{\sqrt{\mathbf{33}}}\mathbf{\text{and}}\\ \mathbf{cosec}\mathbf{\beta }\mathbf{=}\frac{\mathbf{7}}{\sqrt{\mathbf{33}}}\mathbf{\text{,forthetriangle}}\end{array}$

Step-by-step solution

a. Step 1. Given information.

$\begin{array}{l}WehavePisperpendicular,Bisbase,Hishypotenuse.\\ Here,for\angle \alpha ,P=4\\ H=7\end{array}$

a. Step 2. Finding the value of sinα  and   cosβ.

$\begin{array}{l}\sin \alpha =\frac{P}{H}\\ =\frac{4}{7}\\ for\angle \beta ,\\ B=4H=7\\ \cos \beta =\frac{B}{H}\\ \cos \beta =\frac{4}{7}\end{array}$

b. Step 1. Given information.

$\begin{array}{l}WehavePisperpendicular,Bisbase,Hishypotenuse.\\ Here,for\angle \alpha ,P=4\\ H=7\end{array}$

b. Step 2. Finding the value of tanα  and  cotβ

$\begin{array}{l}ByapplyingPythagorasTheorem,weget\\ \begin{array}{c}B=\sqrt{H^2-P^2}\\ \Rightarrow B=\sqrt{7^2-4^2}\\ \Rightarrow B=\sqrt{49-16}\\ \Rightarrow B=\sqrt{33}\\ \tan \alpha =\frac{P}{B}\\ =\frac{4}{\sqrt{33}}\end{array}for\angle \beta ,\\ P=\sqrt{33}\\ \begin{array}{r}B=4\\ H=7\end{array}\\ \begin{array}{c}\cot \beta =\frac{B}{P}\\ =\frac{4}{\sqrt{33}}\end{array}\end{array}$

c. Step 1. Given information.

$\begin{array}{l}WehavePisperpendicular,Bisbase,Hishypotenuse.\\ Here,for\angle \alpha ,P=4\\ H=7\end{array}$

c. Step 2.Finding the value of secα  and  cscβ

By applying Pythagoras Theorem, we get

$B=\sqrt{H^2-P^2}$

$\Rightarrow B=\sqrt{7^2-4^2}$

$\Rightarrow B=\sqrt{49-16}$

$\Rightarrow B=\sqrt{33}$

$\sec \alpha =\frac{H}{B}$

$=\frac{7}{\sqrt{33}}$

$\text{for}\angle \beta ,P=\sqrt{33}$

$B=4$

$\mathrm{cosec}\beta =\frac{H}{P}$

$=\frac{7}{\sqrt{33}}$