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Ch. 5 - Trigonometric Identities
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 5 - Trigonometric IdentitiesProblem 82
Chapter 6, Problem 82

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

Verified step by step guidance
1
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.

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Key Concepts

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

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.
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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.
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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.
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Related Practice
Textbook Question

Advanced methods of trigonometry can be used to find the following exact value.

sin 18° = (√5 - 1)/4

(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.

csc 72°

Textbook Question

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.

x + y = 9, 2x + y = -1

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Textbook Question

Verify that each equation is an identity.

sin³ θ + cos³ θ = (cos θ + sin θ) (1 - cos θ sin θ)

Textbook Question

Advanced methods of trigonometry can be used to find the following exact value.

sin 18° = (√5 - 1)/4

(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.

sin 162°

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views