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### Physics For Scientists & Engineers

Book edition 9th Edition
Author(s) Raymond A. Serway, John W. Jewett
Pages 1624 pages
ISBN 9781133947271

# For the right triangle shown in Figure P1.45, what are (a) the length of the unknown side (b) the tangent of ${\mathbit{\theta }}$, and (c) the sine of ${\mathbit{\varphi }}$?

(a) The length of the unknown side of the right triangle is $6.71\text{\hspace{0.17em}\hspace{0.17em}m}$.

(b) The tangent $\theta$ of the right triangle is $0.894$.

(c) The sine $\varphi$ of the right triangle is $0.745$.

See the step by step solution

## Definition of a right triangle and its formula:

From the three angles of a triangle, if one angle is the right angle (i.e., ${\mathbf{90}}{\mathbf{°}}$). then the triangle is known as a right angle triangle or a right triangle. The three sides of a triangle are named as the base, the altitude, and the hypotenuse. A hypotenuse is an important and the large side in a right triangle.

The formula for determining the sides of a right triangle is given by the Pythagoras Theorem. Based on the Pythagoras Theorem, “the square of the hypotenuse is equal to the sum of the square of the base and the square of the altitude.”

## (a) Calculate the length, tangent, and sine of the right triangle:

Let the unknown length of the right triangle be $x$ and the other known sides be $y$ and $z$.

From the Pythagoras theorem, the unknown is calculated as,

${\text{x}}^{2}={y}^{2}+{z}^{2}$

Replace $9.00\text{\hspace{0.17em}m}$ for localid="1663602823372" $x$ and $6.00\text{\hspace{0.17em}m}$ for $y$ in the above equation.

${\left(9.00\text{\hspace{0.17em}m}\right)}^{2}={\left(6.00\text{\hspace{0.17em}m}\right)}^{2}+{z}^{2}\phantom{\rule{0ex}{0ex}}81{\text{\hspace{0.17em}m}}^{2}=36{\text{\hspace{0.17em}m}}^{2}+{z}^{2}$

$\begin{array}{rcl}{z}^{2}& =& 81{\text{\hspace{0.17em}\hspace{0.17em}m}}^{2}-36{\text{\hspace{0.17em}m}}^{2}\\ & =& 45{\text{\hspace{0.17em}m}}^{2}\\ z& =& 6.71\text{\hspace{0.17em}\hspace{0.17em}m}\end{array}$

Since the significant figures for the known length is three, the result of the unknown length should be in three significant figures (i.e., $6.71\text{\hspace{0.17em}\hspace{0.17em}m}$).

## (b) The tangent of the right triangle:

The tangent $\theta$ is the ratio of the side altitude to the side base of the right triangle.

$\mathrm{tan}\theta =\frac{y}{z}$

Replace $6.00\text{m}$ for $y$ and $6.71\text{m}$ for $z$ in the above equation.

$\begin{array}{rcl}\mathrm{tan}\theta & =& \frac{6.00\text{\hspace{0.17em} m}}{6.71\text{\hspace{0.17em} \hspace{0.17em}m}}\\ & =& 0.894\end{array}$

## (c) The sine of the right triangle:

The $\mathrm{sin}\varphi$ is the ratio of the side base to the side hypotenuse of the right triangle.

$\mathrm{sin}\varphi =\frac{z}{x}$

Replace $9.00\text{\hspace{0.17em}m}$ for $x$ and $\text{6.71\hspace{0.17em}m}$ for $z$ in the above equation.

$\begin{array}{rcl}\mathrm{sin}\varphi & =& \frac{6.71\text{\hspace{0.17em}\hspace{0.17em}m}}{9.00\text{\hspace{0.17em}m}}\\ & =& 0.745\end{array}$

Hence, the length of the unknown side, the tangent $\theta$, and the role="math" localid="1663603918185" $\mathrm{sin}\varphi$ of the right triangle are $6.71\text{\hspace{0.17em}\hspace{0.17em}m}$, $0.894$, and $0.745$ respectively.