4.1 Right triangles

Introduction

The geometry of triangles

Ratios of sides in right triangles

Using a ratio involving sine and cosine

Summary

For each of the following situations, sketch a right triangle that satisfies the given conditions, and then either determine the requested missing information in the triangle or explain why you don't have enough information to determine it. Assume that all angles are being considered in radian measure.

The length of the other leg of a right triangle with hypotenuse of length $1$ and one leg of length $\frac{3}{5}$.

Explain why $\triangle OPQ$ and $\triangle OMN$ are similar triangles.

What is the value of the ratio $\frac{OP}{OM}$? What does this tell you about the ratios $\frac{OQ}{ON}$ and $\frac{PQ}{MN}$?

What is the value of $ON$ in terms of $\theta$? What is the value of $MN$ in terms of $\theta$?

Use your conclusions in (b) and (c) to express the values of $x$ and $y$ in terms of $r$ and $\theta$.

A right triangle with hypotenuse $7$ and one non-right angle of measure $\frac{\pi}{7}$.

A right triangle with non-right angle $\alpha$ that satisfies $\sin(\alpha) = \frac{3}{5}$.

A right triangle where one of the non-right angles has measure $1.2$ and the hypotenuse has length $2.7$.

A right triangle with hypotenuse $13$ and one leg of length $6.5$.

A right triangle with legs of length $5$ and $12$.

A right triangle where one of the non-right angles has measure $\frac{\pi}{5}$ and the leg opposite this angle has length $4$.

Consider a right triangle in which one of the non-right angles is $\beta = \frac{\pi}{5}$ and the leg opposite $\beta$ has length $4$.

Determine (both exactly and approximately) the measures of all of the remaining sides and angles in the triangle.

A person standing $50$ feet away from a streetlight observes that they cast a shadow that is $14$ feet long. If a ray of light from the streetlight to the tip of the person's shadow forms an angle of $27.5^\circ$ with the ground, how tall is the person and how tall is the streetlight? What other information about the situation can you determine?

A person watching a rocket launch uses a laser range-finder to measure the distance from themselves to the rocket. The range-finder also reports the angle at which the finder is being elevated from horizontal. At a certain instant, the range-finder reports that it is elevated at an angle of $17.4^\circ$ from horizontal and that the distance to the rocket is $1650$ meters. How high off the ground is the rocket? Assuming a straight-line vertical path for the rocket that is perpendicular to the earth, how far away was the rocket from the range-finder at the moment it was launched?

A trough is constructed by bending a $4' \times 24'$ rectangular sheet of metal. Two symmetric folds $2$ feet apart are made parallel to the longest side of the rectangle so that the trough has cross-sections in the shape of a trapezoid, as pictured in Figure. Determine a formula for $V(\theta)$, the volume of the trough as a function of $\theta$.