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Chapter 16: Problem 10

If the short leg of a right triangle is 5 units long and the long leg is 7 units long, what is the angle opposite the short leg, in degrees? a) \(26.3\) b) \(28.9\) c) \(31.2\) d) \(35.5\)

Short Answer

Expert verified
a) \(25.5\) b) \(30.5\) c) \(34.5\) d) \(35.5\) e) \(37.5\) Answer: d) \(35.5\)

Step by step solution

01

Identify the given values and the desired angle

In this right triangle, the short leg's length is 5 units, the long leg's length is 7 units, and we are asked to find the angle opposite the short leg, let's call this angle θ.
02

Set up the tangent function

For a right triangle, the tangent function is defined by tan(θ) = opposite leg/adjacent leg. In this case, the short leg (opposite) is 5 units, and the long leg (adjacent) is 7 units. Thus: tan(θ) = 5/7
03

Solve for the angle θ, in radians

In order to find the angle θ, we use the inverse tangent function (arctan) to the previous equation: θ = arctan(5/7)
04

Convert the angle from radians to degrees

Once we get the angle θ in radians, it is necessary to convert it to degrees since the answer options are in degrees. To convert from radians to degrees, we will multiply the angle by the conversion factor (180/π): θ (degrees) = θ (radians) × (180/π)
05

Find the approximate value of the angle θ and choose the correct answer

By using a calculator to compute the angle in degrees, we obtain: θ (degrees) ≈ 35.54 Rounded to one decimal place, the angle is 35.5 degrees, which corresponds to answer choice d) \(35.5\).

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Most popular questions from this chapter

A \(100-\mathrm{m}\)-long track is to be built \(50 \mathrm{~m}\) wide. If it is to be elliptical, what equation could describe it if the 100 -m length is along the \(x\)-axis? a) \(50 x^{2}+100 y^{2}=1000\) c) \(4 x^{2}+y^{2}=2500\) b) \(2 x^{2}+y^{2}=250\) d) \(x^{2}+2 y^{2}=250\)

Evaluate \(\int_{0}^{2}\left(e^{x}+\sin x\right) d x\) a) \(7.81\) b) \(6.21\) c) \(5.92\) d) \(5.61\)

Derive an expression \(\int x \cos x d x\) a) \(x \cos x-\sin x+C\) c) \(x \sin x-\cos x+C\) b) \(x \sin x+\cos x+C\) d) \(x \cos x+\sin x+C\)

The shortest distance from the line \(3 x-4 y=3\) to the point \((6,8)\) is a) \(4.8\) b) \(4.2\) c) \(3.8\) d) \(3.4\)

A triangle has sides of length 2,3 , and 4 . What angle, in radians, is opposite the side of length 3 ? a) \(0.55\) b) \(0.61\) c) \(0.76\) d) \(0.81\)
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