Ch. 1 - Trigonometric Functions

Lial12th EditionTrigonometryISBN: 9780136552161당신이 사용하는 게 아니라요?교과서 변경

Lial 12th Edition

Ch. 1 - Trigonometric Functions

문제 67

2장, 문제 67

In each figure, there are two similar triangles. Find the unknown measurement. Give approximations to the nearest tenth.

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Identify the corresponding sides of the two similar triangles. Since the triangles are similar, their corresponding sides are proportional.

Set up a proportion using the lengths of the known sides from both triangles. For example, if side \(a\) in the smaller triangle corresponds to side \(A\) in the larger triangle, and side \(b\) corresponds to side \(B\), then write the proportion as \(\frac{a}{A} = \frac{b}{B}\).

Substitute the known side lengths into the proportion, leaving the unknown measurement as a variable (e.g., \(x\)).

Solve the proportion equation for the unknown variable by cross-multiplying and isolating the variable on one side of the equation.

Calculate the value of the unknown measurement and round your answer to the nearest tenth as requested.

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주요 개념

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Similarity of Triangles

Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional. This means one triangle is a scaled version of the other, which allows us to set up ratios between corresponding sides to find unknown lengths.

Proportionality of Corresponding Sides

In similar triangles, the ratios of the lengths of corresponding sides are equal. This property enables solving for unknown side lengths by setting up and solving proportion equations based on known side measurements.

Rounding and Approximation

When calculating unknown measurements, the results may be irrational or decimal numbers. Rounding to the nearest tenth means adjusting the value to one decimal place, which simplifies the answer while maintaining reasonable accuracy.

관련 실천

교과서 질문

Determine whether each statement is possible or impossible. See Example 4. cot θ = ―6

917

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교과서 질문

Convert each angle measure to degrees, minutes, and seconds. If applicable, round to the nearest second. See Example 4(b). 174.255°

629

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교과서 질문

Use identities to solve each of the following. Rationalize denominators when applicable. See Examples 5–7. Find csc θ , given that cot θ = ―1/2 and θ is in quadrant IV.

701

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교과서 질문

Find the indicated function value. If it is undefined, say so. See Example 4. sin 90°

670

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교과서 질문

Solve each problem. Solar Eclipse on Earth The sun has a diameter of about 865,000 mi with a maximum distance from Earth's surface of about 94,500,000 mi. The moon has a smaller diameter of 2159 mi. For a total solar eclipse to occur, the moon must pass between Earth and the sun. The moon must also be close enough to Earth for the moon's umbra (shadow) to reach the surface of Earth. (Data from Karttunen, H., P. Kröger, H. Oja, M. Putannen, and K. Donners, Editors, Fundamental Astronomy, Fourth Edition, Springer-Verlag.) a. Calculate the maximum distance, to the nearest thousand miles, that the moon can be from Earth and still have a total solar eclipse occur. (Hint: Use similar triangles.)

804

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교과서 질문

Find the indicated function value. If it is undefined, say so. See Example 4. sec 180°

707

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