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Chapter 19: Problem 1881

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) \((\pi / 3)\)

Short Answer

Expert verified
\(B = \frac{\pi}{6}\)

Step by step solution

01

Recall the sine rule

The sine rule for any triangle states that: \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\] where a, b, and c are the side lengths of the triangle, and A, B, and C are the angles opposite to those sides respectively.
02

Write down the given information

The given information is: a = 2 b = 3 \(\sin A = \frac{1}{3}\)
03

Apply the sine rule to find sine of angle B

Using the sine rule, we have: \[\frac{a}{\sin A} = \frac{b}{\sin B}\] Plug in the given values: \[\frac{2}{\frac{1}{3}} = \frac{3}{\sin B}\] Simplifying the equation, we get, \[\sin B = \frac{9}{2}\]
04

Correct the value of sine of angle B

Since the value of sine cannot exceed 1, the value of sin B above is incorrect.
05

Revisit the sine rule and note the alternative form

The sine rule can also be written as follows: \[\frac{\sin A}{a} = \frac{\sin B}{b}\]
06

Apply the alternative form of the sine rule

Using the alternative form of the sine rule, we have: \[\frac{\sin A}{a} = \frac{\sin B}{b}\] Plug in the given values: \[\frac{\frac{1}{3}}{2} = \frac{\sin B}{3}\]
07

Solve for sine of angle B

Multiply both sides by 3 to solve for sin B: \[\sin B = 3 \cdot \frac{\frac{1}{3}}{2} = \frac{1}{2}\]
08

Find angle B using the inverse sine function

Since \(\sin{B} = \frac{1}{2}\), we find angle B by using the inverse sine function: \[B = \sin^{-1} \left( \frac{1}{2} \right)\] From here, we find that angle B is equal to \(\frac{\pi}{6}\) (option b).

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Most popular questions from this chapter

If \(x=\tan 10^{\circ}\), then \(\tan 70^{\circ}=\) (a) \(\left[2 x /\left(1-x^{2}\right)\right]\) (b) \(\left[\left(1-x^{2}\right) / 2 x\right]\) (c) \(7 x\) (d) \(2 \mathrm{x}\)

\({ }^{\infty} \sum_{r=1} \tan ^{-1}\left(1 / 2 r^{2}\right)=\) (a) \((\pi / 4)\) (b) \((\pi / 2)\) (c) \(\tan ^{-1}(\mathrm{n})-(\pi / 4)\) (d) \(\tan ^{-1}(n+1)-(\pi / 4)\)

The minimum value of \(125 \tan ^{2} \theta+5 \cot ^{2} \theta\) is (a) 5 (b) 25 (c) 125 (d) 50

If \(A=\cos ^{4} \theta+\sin ^{2} \theta, \forall \theta \in R\) then \(A\) lies in the interval (a) \([1,2]\) (b) \([(3 / 4), 1]\) (c) \([(13 / 16), 1]\) (d) \([(3 / 4),(13 / 16)]\)

Right circular cone has a height \(40 \mathrm{~cm}\) and its semi vertical angle is \(45^{\circ}\) then radius of its base circle is (a) \(40 \mathrm{~cm}\) (b) \(80 \mathrm{~cm}\) (c) \([(40 \sqrt{3}) / 2] \mathrm{cm}\) (d) \(20 \mathrm{~cm}\)
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