Textbook Question
In Exercises 55–58, use a graph to solve each equation for -2π ≤ x ≤ 2π. tan x = -1
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2. Trigonometric Functions on Right Triangles
Solving Right Triangles
2:37 minutesProblem 30
Textbook Question
In Exercises 30–32, find the measure of the side of the right triangle whose length is designated by a lowercase letter. Round answers to the nearest whole number.
Verified step by step guidance1
Identify the sides of the right triangle relative to the given angle of 48° at vertex P. The side adjacent to the angle is PR = 700 m, the side opposite the angle is QR, and the hypotenuse is PQ.
Use the trigonometric ratios to relate the sides to the angle. For example, to find the hypotenuse PQ, use the cosine function: \(\cos(48^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{700}{PQ}\).
Rearrange the equation to solve for the hypotenuse PQ: \(PQ = \frac{700}{\cos(48^\circ)}\).
To find the opposite side QR, use the sine function: \(\sin(48^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{QR}{PQ}\).
Rearrange the sine equation to solve for QR: \(QR = PQ \times \sin(48^\circ)\). Substitute the expression for PQ from step 3 to find QR in terms of known values.
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Video duration:
2mKey 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 problems involving right triangles.
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30-60-90 Triangles
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 allow calculation of unknown side lengths or angles when one side and one angle are known.
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Using Given Angle and Side to Find Unknown Sides
Given an angle and one side length in a right triangle, you can use trigonometric ratios to find the other sides. For example, with angle 48° and adjacent side 700 m, cosine can find the hypotenuse, and tangent can find the opposite side. Rounding to the nearest whole number is often required.
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