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Chapter 19: Problem 1826

If \(\sin \left(120^{\circ}-\alpha\right)=\sin \left(120^{\circ}-\beta\right)\) and \(0<\alpha, \beta<\pi\) then all values of \(\alpha, \beta\) are given by (a) \(\alpha+\beta=(\pi / 3)\) (b) \(\alpha=\beta\) (c) \(\alpha=\beta\) or \(\alpha+\beta=(\pi / 3)\) (d) \(a+\beta=0\)

Short Answer

Expert verified
The short answer is: \(\alpha = \beta\) or \(\alpha + \beta = \pi / 3\), which corresponds to choice (c).

Step by step solution

01

Rewrite the Given Equation using Angle Identities

Since we want to find a relationship between the angles \(\alpha\) and \(\beta\), we can rewrite the given equation by using the angle subtraction identity for sine: \[\sin(a - b) = \sin(a) \cos(b) - \cos(a) \sin(b).\] Applying this identity to both sides of our equation, we have: \[\sin(120^\circ - \alpha) = \sin(120^\circ) \cos(\alpha) - \cos(120^\circ) \sin(\alpha) \] and \[\sin(120^\circ - \beta) = \sin(120^\circ) \cos(\beta) - \cos(120^\circ) \sin(\beta).\] Since the left-hand sides of these equations are equal, we can equate the right-hand sides: \[\sin(120^\circ) \cos(\alpha) - \cos(120^\circ) \sin(\alpha) = \sin(120^\circ) \cos(\beta) - \cos(120^\circ) \sin(\beta).\]
02

Simplify the Equation

We can simplify the equation by factoring out the common terms \(\sin(120^\circ)\) and \(\cos(120^\circ)\): \[\sin(120^\circ) (\cos(\alpha) - \cos(\beta)) - \cos(120^\circ) (\sin(\alpha) - \sin(\beta)) = 0.\] Now we can use the angle addition formulas for \(\sin\) and \(\cos\) to rewrite the terms in parentheses: \[\sin(180^\circ - \beta) = \sin(\beta),\] \[\cos(180^\circ - \beta) = -\cos(\beta),\] thus, \[\sin(120^\circ) (\cos(\alpha) + \cos(\beta)) - \cos(120^\circ) (\sin(\alpha) - \sin(\beta)) = 0.\]
03

Solve for the Relationship between \(\alpha\) and \(\beta\)

Since \(\sin(120^\circ)\neq0\) and \(\cos(120^\circ)\neq0\), we must have either: 1) \(\cos(\alpha) + \cos(\beta) = 0\), or 2) \(\sin(\alpha) - \sin(\beta) = 0\). For case 1, we find that \(\alpha+\beta=180^{\circ}=\pi / 3\). For case 2, we find that \(\alpha=\beta\). Since these are the only two possibilities, the correct answer is \[\alpha=\beta \,\,\text{or}\,\, \alpha+\beta=\pi / 3,\] which matches choice (c).

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Most popular questions from this chapter

The angle of elevation of a parachute measured from a point at a height \(60 \mathrm{~m}\) from the surface of a lake is \(30^{\circ}\) and the angle of depression of reflection of parachute seen in the lake from the same point is \(60^{\circ}\). Then height of the parachute from the surface of a lake is (a) \(120 \mathrm{~m}\) (b) \(60 \mathrm{~m}\) (c) \(90 \mathrm{~m}\) (d) \(150 \mathrm{~m}\)

If \(\Delta \mathrm{ABC}, \mathrm{a}=2, \mathrm{~b}=3\) and \(\sin \mathrm{A}=(1 / 3)\), then \(\mathrm{B}=\) (a) \((\pi / 4)\) (b) \((\pi / 6)\) (c) \((\pi / 2)\) (d) \((\pi / 3)\)

There is a bridge of the length \(h\) on a valley. The angle of depression of a temple lying in a valley from two ends of a bridge are \(\alpha\) and \(\beta\), then the height of the bridge from top of the temple \(=\) (a) \([(h \tan \alpha \tan \beta) /(\tan \alpha-\tan \beta)]\) (b) \([(h \tan \alpha \tan \beta) /(\tan \alpha+\tan \beta)]\) (c) \([(\tan \alpha \tan \beta) /\\{h(\tan \alpha-\tan \beta)\\}]\) (d) \([\\{h(\tan \alpha+\tan \beta)\\} /(\tan \alpha \tan \beta)]\)

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If \(\cos x+\cos y=0\) and \(\sin x+\sin y=0\) then \(\cos (x-y)=\) (a) 1 (b) \((1 / 2)\) (c) \(-1\) (d) \(-(1 / 2)\)
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