Chapter 7: Q. 102 (page 498)

If $x = 2 \tan θ$ , express $\cos (2 θ)$ as a function ofx.

Short Answer

Expert verified

On expressing $\cos (2 θ)$ as a function of xis $\cos (2 θ) = \frac{4 - x^{2}}{4 + x^{2}}$ .

Step by step solution

Step 1. Given information.

Consider the given question,

$x = 2 \tan θ$

From the double-angle formula,

$\cos (2 θ) = 2 \cos^{2} θ - 1$

From inverse trigonometry,

$θ = \tan^{- 1} \frac{x}{2} = \sin^{- 1} (\frac{x}{\sqrt{4 + x^{2}}}) = c o s^{- 1} (\frac{2}{\sqrt{4 + x^{2}}})$

Step 2. Use inverse trigonometry and solve the equation.

From inverse trigonometry,

$θ = \tan^{- 1} \frac{x}{2}$

Then,

$θ = \tan^{- 1} \frac{x}{2} \cos (2 θ) = 2 \cos^{2} (\tan^{- 1} \frac{x}{2}) - 1 \cos (2 θ) = 2 \cos^{2} (\cos^{- 1} (\frac{2}{\sqrt{4 + x^{2}}})) - 1 \cos (2 θ) = 2 {(\frac{2}{\sqrt{4 + x^{2}}})}^{2} - 1 \cos (2 θ) = \frac{4 - x^{2}}{4 + x^{2}}$

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If $x = 2 \tan θ$ , express $\cos (2 θ)$ as a function ofx.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use inverse trigonometry and solve the equation.

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If x=2tanθ, express cos2θas a function ofx.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use inverse trigonometry and solve the equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Decision Maths

Mechanics Maths

Geometry

Discrete Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

If $x = 2 \tan θ$ , express $\cos (2 θ)$ as a function ofx.