Chapter 3: Q26P (page 123)

In(6.14), let $θ = ϕ = ττ / 2$ and verify the result numerically.

Short Answer

Expert verified

The relation $(\begin{matrix} \cos ϕ & - \sin ϕ \\ \sin ϕ & \cos ϕ \end{matrix}) (\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}) = (\begin{matrix} \cos (θ + ϕ) & - \sin (θ + ϕ) \\ \sin (θ + ϕ) & \cos (θ + ϕ) \end{matrix})$ is verified numerically for and its numerical value is $(\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix})$ .

Step by step solution

The trigonometric values of sine and cosine function for different angles:

The numerical values of the sine and cosine function at $\frac{π}{2}$ , and $π$ are

$s i n (\frac{π}{2}) = 1 c o s (\frac{π}{2}) = 0 s i n (π) = 0 c o s (π) = - 1$

Given parameters:

The result (6.14) is $(\begin{matrix} \cos ϕ & - \sin ϕ \\ \sin ϕ & \cos ϕ \end{matrix}) (\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}) = (\begin{matrix} \cos (θ + ϕ) & - \sin (θ + ϕ) \\ \sin (θ + ϕ) & \cos (θ + ϕ) \end{matrix})$ .

It needs to be verified for $θ = ϕ = π / 2$ .

Finding the numerical values of the left-hand side and the right-hand side of the relation:

Substitute into the left-hand side of the relation $(6.14)$ and evaluate the result.

$(\begin{matrix} \cos \frac{π}{2} & - \sin \frac{π}{2} \\ \sin \frac{π}{2} & \cos \frac{π}{2} \end{matrix}) (\begin{matrix} \cos \frac{π}{2} & - \sin \frac{π}{2} \\ \sin \frac{π}{2} & \cos \frac{π}{2} \end{matrix}) = (\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}) (\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix})$

Multiply the matrices.

Substitute $θ = ϕ = π / 2$ into the right-hand side of the relation (6.14).

role="math" localid="1658989214239" $(\begin{matrix} \cos (\frac{π}{2} + \frac{π}{2}) & - \sin (\frac{π}{2} + \frac{π}{2}) \\ \sin (\frac{π}{2} + \frac{π}{2}) & \cos (\frac{π}{2} + \frac{π}{2}) \end{matrix}) = (\begin{matrix} \cos (π) & - \sin (π) \\ \sin (π) & \cos (π) \end{matrix})$

Evaluate the sine and cosine function.

$(\begin{matrix} \cos (π) & - \sin (π) \\ \sin (π) & \cos (π) \end{matrix}) = (\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix})$

Hence, the relation (6.14) is verified numerically

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In(6.14), let $θ = ϕ = ττ / 2$ and verify the result numerically.

Short Answer

Step by step solution

The trigonometric values of sine and cosine function for different angles:

Given parameters:

Finding the numerical values of the left-hand side and the right-hand side of the relation:

One App. One Place for Learning.

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In(6.14), let θ=ϕ=ττ/2 and verify the result numerically.

Short Answer

Step by step solution

The trigonometric values of sine and cosine function for different angles:

Given parameters:

Finding the numerical values of the left-hand side and the right-hand side of the relation:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Geometrical and Physical Optics

Linear Momentum

Electricity and Magnetism

Fundamentals of Physics

Magnetism and Electromagnetic Induction

Kinematics Physics

Study anywhere. Anytime. Across all devices.

In(6.14), let $θ = ϕ = ττ / 2$ and verify the result numerically.