Use the result from Exercise 80 to find the acute angle between each pair of lines. (Note that the tangent of the angle will be positive.) Use a calculator, and round to the nearest tenth of a degree.
5x - 2y + 4 = 0, 3x + 5y = 6

검증된 단계별 안내

Rewrite each line in slope-intercept form \(y = mx + b\) to identify their slopes. For the first line \(5x - 2y + 4 = 0\), solve for \(y\) to find its slope \(m_1\).

Similarly, rewrite the second line \(3x + 5y = 6\) in slope-intercept form to find its slope \(m_2\).

Use the formula for the tangent of the angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\): \(\tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|\) This formula gives the absolute value of the tangent of the angle between the lines.

Calculate the value of \(\tan(\theta)\) using the slopes found in steps 1 and 2. Since the problem states the tangent is positive, take the positive value.

Use a calculator to find the angle \(\theta\) by taking the arctangent (inverse tangent) of the value from step 4: \(\theta = \arctan\left( \tan(\theta) \right)\) Round the result to the nearest tenth of a degree to get the acute angle between the lines.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Angle Between Two Lines

The angle between two lines in a plane can be found using the slopes of the lines. If m1 and m2 are the slopes, the tangent of the angle θ between them is given by |(m1 - m2) / (1 + m1*m2)|. This formula helps determine the acute angle formed where the lines intersect.

Finding the Slope from a Line Equation

To find the slope of a line given in standard form Ax + By + C = 0, rearrange it into slope-intercept form y = mx + b. The slope m is then -A/B. This step is essential to apply the angle formula between two lines.

Using a Calculator to Find Angles from Tangent Values

Once the tangent of the angle is calculated, use the inverse tangent function (arctan or tan⁻¹) on a calculator to find the angle in degrees. Since the problem specifies the tangent is positive, the angle found will be acute and should be rounded to the nearest tenth of a degree.

Use the result from Exercise 80 to find the acute angle between each pair of lines. (Note that the tangent of the angle will be positive.) Use a calculator, and round to the nearest tenth of a degree.5x - 2y + 4 = 0, 3x + 5y = 6