Chapter 7: Q. 112 (page 487)

If $\tan θ = a \tan \frac{θ}{3}$ , express $\tan \frac{θ}{3}$ in terms of a.

Short Answer

Expert verified

On expressing $\tan \frac{θ}{3}$ in terms of a, we get $0, \sqrt{\frac{a - 3}{3 a - 1}}, - \sqrt{\frac{a - 3}{3 a - 1}}$ .

Step by step solution

Step 1. Given information.

Consider the given question,

$\tan θ = a \tan \frac{θ}{3}$

We know localid="1646426304523" $\tan (3 θ) = \frac{3 \tan θ - \tan^{3} θ}{1 - 3 \tan^{2} θ}$ . Then,

localid="1646427644279" role="math" $\frac{3 \tan \frac{θ}{3} - \tan^{3} \frac{θ}{3}}{1 - 3 \tan^{2} \frac{θ}{3}} = a \tan \frac{θ}{3} 3 \tan \frac{θ}{3} - \tan^{3} \frac{θ}{3} = a \tan \frac{θ}{3} (1 - 3 \tan^{2} \frac{θ}{3})$

Step 2. Simply the equation.

Continuing the above equation,

$3 \tan \frac{θ}{3} - \tan^{3} \frac{θ}{3} = a \tan \frac{θ}{3} - 3 a \tan^{3} \frac{θ}{3} (3 a - 1) \tan^{3} \frac{θ}{3} + (3 - a) \tan \frac{θ}{3} = 0 \tan \frac{θ}{3} = 0, \pm \sqrt{\frac{a - 3}{3 a - 1}}$

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If $\tan θ = a \tan \frac{θ}{3}$ , express $\tan \frac{θ}{3}$ in terms of a.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Simply the equation.

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Short Answer

Step by step solution

Step 1. Given information.

Step 2. Simply the equation.

One App. One Place for Learning.

Most popular questions from this chapter

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If $\tan θ = a \tan \frac{θ}{3}$ , express $\tan \frac{θ}{3}$ in terms of a.