Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Problem 3

Chapter 2, Problem 3

Solve the right triangle shown in the figure. Round lengths to two decimal places and express angles to the nearest tenth of a degree. A = 52.6°, c = 54

Verified step by step guidance

Identify the given elements in the right triangle: angle \(A = 52.6^\circ\) at vertex \(Q\), and hypotenuse \(c = 54\) (which corresponds to side \(r\) in the diagram).

Since the triangle is right-angled at \(R\), use the fact that the sum of angles in a triangle is \(180^\circ\). Calculate the other non-right angle \(B\) using \(B = 90^\circ - A\).

Use the sine and cosine definitions to find the legs \(p\) and \(q\): \(p = c \times \sin(A)\) and \(q = c \times \cos(A)\), where \(p\) is opposite angle \(A\) and \(q\) is adjacent to angle \(A\).

Substitute the known values into the formulas: \(p = 54 \times \sin(52.6^\circ)\) and \(q = 54 \times \cos(52.6^\circ)\).

Calculate the values of \(p\) and \(q\) using a calculator, then round the lengths to two decimal places and the angles to the nearest tenth of a degree as required.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Right Triangle Properties

A right triangle has one angle of 90 degrees, and the side opposite this angle is the hypotenuse, the longest side. The other two sides are called legs. Understanding the relationship between these sides and angles is essential for solving the triangle.

Trigonometric Ratios (Sine, Cosine, Tangent)

Trigonometric ratios relate the angles of a right triangle to the ratios of its sides. Sine is opposite/hypotenuse, cosine is adjacent/hypotenuse, and tangent is opposite/adjacent. These ratios help find unknown sides or angles when some measurements are known.

Angle and Side Calculation Using Trigonometry

Given an angle and a side, you can use trigonometric functions to calculate the other sides and angles. For example, using sine or cosine with the known angle and hypotenuse allows finding the legs, and the remaining angle is found by subtracting from 90°.

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