In triangle P Q R shown in Fig. 12.14, find the lengths of P...

In triangle $P Q R$ shown in Fig. 12.14, find the lengths of $P Q$ and $P R$. $\tan 38^{\circ}=\frac{P Q}{Q R}=\frac{P Q}{7.5}$ hence $\quad P Q=7.5 \tan 38^{\circ}=7.5(0.7813)$ $=5.860 \mathrm{~cm}$ $\cos 38^{\circ}=\frac{Q R}{P R}=\frac{7.5}{P R}$ hence $\quad P R=\frac{7.5}{\cos 38^{\circ}}=\frac{7.5}{0.7880}=9.518 \mathrm{~cm}$

In triangle $P Q R$ shown in Fig. 12.14, find the lengths of $P Q$ and $P R$. $\tan 38^{\circ}=\frac{P Q}{Q R}=\frac{P Q}{7.5}$
hence $\quad P Q=7.5 \tan 38^{\circ}=7.5(0.7813)$
$=5.860 \mathrm{~cm}$
$\cos 38^{\circ}=\frac{Q R}{P R}=\frac{7.5}{P R}$
hence $\quad P R=\frac{7.5}{\cos 38^{\circ}}=\frac{7.5}{0.7880}=9.518 \mathrm{~cm}$

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Key Concepts

Cosine Function

The cosine function in a right triangle is defined as the ratio of the length of the side adjacent to an angle to the length of the hypotenuse. This ratio is used to determine an unknown side when the hypotenuse or the adjacent side is known, and it plays a crucial role in linking the angle measures with side lengths.

Tangent Function

The tangent function in a right triangle is defined as the ratio of the length of the side opposite to an angle to the length of the side adjacent to that angle. This function is particularly useful when one of these two sides is known, and the corresponding angle is given, allowing for the calculation of the unknown side.

Trigonometric Ratios

Trigonometric ratios are relationships between the sides of a right triangle relative to its acute angles. They include the sine, cosine, and tangent functions, each defined as a ratio of two specific sides, and are essential tools for solving problems involving measurements in right triangles.

Right Triangle

A right triangle is a triangle that has one angle measuring exactly 90°. This type of triangle allows the use of specific trigonometric ratios—sine, cosine, and tangent—which relate the angles to the lengths of the sides, making it fundamental in solving for unknown side lengths or angles.

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