Chapter 6: Q10E (page 326)

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.
${sinx, cosx, tanx}$ on $(- \frac{π}{2}, \frac{π}{2})$

Short Answer

Expert verified

The function $\{sinx, cosx, tanx\}$ is linearly independent on $(- \frac{π}{2}, \frac{π}{2})$ .

Step by step solution

Step 1:Using the concept of Wronskian

The given function is $\{sinx, cosx, tanx\} .$

Apply the concept of Wronskian,

$W [f_{1}, f_{2}, \dots, f_{n}] = |\begin{matrix} f_{1} (x) & f_{2} (x) & \dots & f_{n} (x) \\ f_{1}' (x) & f_{2}' (x) & \dots & f_{n}' (x) \\ ⋮ & ⋮ & ⋮ \\ {f_{1}}^{n - 1} (x) & {f_{2}}^{n - 1} (x) & \dots & {f_{n}}^{n - 1} (x) \end{matrix}|$

Therefore,

$W [\sin x, \cos x, \tan x] = |\begin{matrix} \sin x & \cos x & \tan x \\ \cos x & - \sin x & s e c^{2} x \\ - \sin x & - \cos x & 2 s e c^{2} x \tan x \end{matrix}|$

Solve the above equation,

$W [sinx, cosx, tanx] = sinx (- 2 {sinxsec}^{2} xtanx + {cosxsec}^{2} x) - cosx (2 {cosxsec}^{2} xtanx + {sinxsec}^{2} x) + tanx (- \cos^{2} x - \sin^{2} x) = - 2 \sin^{2} {xsec}^{2} xtanx + {sinxcosxsec}^{2} x - 2 \cos^{2} {xsec}^{2} xtanx - {sinxcosxsec}^{2} x - tanx = - 2 \sec^{2} xtanx - tanx = - tanx (2 \sec^{2} x - 1) = - tanx (2 \tan^{2} x + 2 - 1) = - tanx (2 \tan^{2} x + 1)$

Step 2:Check the linearly independent or dependent

The above function is not equal to zero $(\forall x) .$

Therefore, the function $\{sinx, cosx, tanx\}$ is linearly independent on $(- \frac{π}{2}, \frac{π}{2})$ .

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Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.
${sinx, cosx, tanx}$ on $(- \frac{π}{2}, \frac{π}{2})$

Short Answer

Step by step solution

Step 1:Using the concept of Wronskian

Step 2:Check the linearly independent or dependent

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Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.{sinx, cosx, tanx}on(-π2, π2)

Short Answer

Step by step solution

Step 1:Using the concept of Wronskian

Step 2:Check the linearly independent or dependent

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Geometry

Mechanics Maths

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Study anywhere. Anytime. Across all devices.

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.
${sinx, cosx, tanx}$ on $(- \frac{π}{2}, \frac{π}{2})$