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 31

Chapter 2, Problem 31

In Exercises 29–36, find the length x to the nearest whole unit.

Verified step by step guidance

Identify the two right triangles in the diagram. The larger triangle has a base of 1600 units and an unknown height x. The angles given are 66° and 24°, which are complementary to the right angle in each triangle.

Use the tangent function, which relates the opposite side to the adjacent side in a right triangle. For the larger triangle with angle 66°, write the equation: \(\tan(66^\circ) = \frac{x}{1600}\).

Solve for x in the equation from step 2: \(x = 1600 \times \tan(66^\circ)\).

For the smaller triangle, note that the height is the difference between the two heights formed by the two triangles. Use the tangent function for the 24° angle: \(\tan(24^\circ) = \frac{h}{1600}\), where h is the height of the smaller triangle.

Calculate the height h from the smaller triangle: \(h = 1600 \times \tan(24^\circ)\). Then, subtract h from the height x of the larger triangle to find the length of the segment labeled x in the diagram.

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Key Concepts

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

Right Triangle Trigonometry

Right triangle trigonometry involves relationships between the angles and sides of a right triangle. The primary trigonometric ratios—sine, cosine, and tangent—relate an angle to the ratios of two sides. These ratios are essential for finding unknown side lengths or angles when some measurements are known.

Angle Sum Property of Triangles

The angle sum property states that the sum of the interior angles of any triangle is 180°. In this problem, knowing two angles (66° and 24°) helps confirm the triangle's structure and can assist in identifying complementary or supplementary angles needed for calculations.

Using Trigonometric Ratios to Find Lengths

To find an unknown side length like x, use trigonometric ratios with the given angles and known side lengths. For example, tangent relates the opposite side to the adjacent side, allowing calculation of height x when the base and angle are known. This approach is key to solving the problem.

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Related Practice

Textbook Question

In Exercises 31–34, determine the amplitude of each function. Then graph the function and y = cos x in the same rectangular coordinate system for 0 ≤ x ≤ 2π. y = 2 cos x

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In Exercises 29–44, graph two periods of the given cosecant or secant function. y = 1/2 csc x/2

Textbook Question

In Exercises 29–36, find the length x to the nearest whole unit.

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In Exercises 29–51, find the exact value of each expression. Do not use a calculator. _ sin⁻¹ (− √3/2)

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