Chapter 2: Q27P (page 67)

Using the fact that a complex equation is really two real equations, find the double angle formulas $(f o r s i n 2 θ, c o s 2 θ)$ by using equation 10.2.

Short Answer

Expert verified

The double angle formula for $(\sin 2 θ, \cos 2 θ)$ is $\cos 2 θ = \cos^{2} θ - \sin^{2} θ$

Step by step solution

Given Information

To find the double angle formulas for $(\sin 2 θ, \cos 2 θ)$ using equation 10.2.

Definition of the complex number

Complex numbers are represented in terms of real numbers and imaginary numbers; a complex can be written in the form of:

$z = a + i b$

Here a and b are real numbers, and i is the imaginary number which is known as iota, whose value is $\sqrt{- 1}$ .

Finding an expression for and

Exponential form for z ;

$U = e^{{(i θ)}^{2}} = e^{(2 i θ)} = \cos 2 θ + i \sin 2 θ$

From the above, it can further have written as,

$U = e^{{(i θ)}^{2}} = {[\cos θ + \sin θ]}^{2}$

……. (1)

Simplifying the expression (1), we get,

role="math" localid="1658740750975" $U = \cos^{2} (θ) - \sin^{2} (θ) + 2 \sin (θ) \cos (θ) i$ ….… (2)

From (1) and (2) RE = RE and lm = lm therefore;

$\cos 2 θ = \cos^{2} θ - \sin^{2} θ \sin 2 θ = 2 \sin θ \cos θ$

Hence the formula will be, $\cos 2 θ = \cos^{2} (θ) - \sin^{2} (θ), \sin (θ) = 2 \sin θ \cos θ$ .

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Using the fact that a complex equation is really two real equations, find the double angle formulas $(f o r s i n 2 θ, c o s 2 θ)$ by using equation 10.2.

Short Answer

Step by step solution

Given Information

Definition of the complex number

Finding an expression for and

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Using the fact that a complex equation is really two real equations, find the double angle formulas (forsin2θ,cos2θ)by using equation 10.2.

Short Answer

Step by step solution

Given Information

Definition of the complex number

Finding an expression for and

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Astrophysics

Turning Points in Physics

Magnetism and Electromagnetic Induction

Mechanics and Materials

Quantum Physics

Electrostatics

Study anywhere. Anytime. Across all devices.

Using the fact that a complex equation is really two real equations, find the double angle formulas $(f o r s i n 2 θ, c o s 2 θ)$ by using equation 10.2.