Linear Algebra Examples

[\begin{matrix} sin (t h e t a) & - 1 - 1 & sin (t h e t a) \end{matrix}]

$[\begin{array}{c} \sin (t h e t a) & - 1 \\ - 1 & \sin (t h e t a) \end{array}]$

Step 1

Multiply

t

$t$ by

t

$t$ by adding the exponents.

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Step 1.1

Move

t

$t$ .

[\begin{matrix} sin (t \cdot t h e a) & - 1 - 1 & sin (t h e t a) \end{matrix}]

$[\begin{array}{c} \sin (t \cdot t h e a) & - 1 \\ - 1 & \sin (t h e t a) \end{array}]$

Step 1.2

Multiply

t

$t$ by

t

$t$ .

[\begin{matrix} sin (t^{2} h e a) & - 1 - 1 & sin (t h e t a) \end{matrix}]

$[\begin{array}{c} \sin (t^{2} h e a) & - 1 \\ - 1 & \sin (t h e t a) \end{array}]$

[\begin{matrix} sin (t^{2} h e a) & - 1 - 1 & sin (t h e t a) \end{matrix}]

$[\begin{array}{c} \sin (t^{2} h e a) & - 1 \\ - 1 & \sin (t h e t a) \end{array}]$

Step 2

Multiply

t

$t$ by

t

$t$ by adding the exponents.

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Step 2.1

Move

t

$t$ .

[\begin{matrix} sin (t^{2} h e a) & - 1 - 1 & sin (t \cdot t h e a) \end{matrix}]

$[\begin{array}{c} \sin (t^{2} h e a) & - 1 \\ - 1 & \sin (t \cdot t h e a) \end{array}]$

Step 2.2

Multiply

t

$t$ by

t

$t$ .

[\begin{matrix} sin (t^{2} h e a) & - 1 - 1 & sin (t^{2} h e a) \end{matrix}]

$[\begin{array}{c} \sin (t^{2} h e a) & - 1 \\ - 1 & \sin (t^{2} h e a) \end{array}]$

[\begin{matrix} sin (t^{2} h e a) & - 1 - 1 & sin (t^{2} h e a) \end{matrix}]

$[\begin{array}{c} \sin (t^{2} h e a) & - 1 \\ - 1 & \sin (t^{2} h e a) \end{array}]$

Step 3

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

sin (t^{2} h e a) sin (t^{2} h e a) - - - 1

$\sin (t^{2} h e a) \sin (t^{2} h e a) - - - 1$

Step 4

Simplify the determinant.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply

sin (t^{2} h e a) sin (t^{2} h e a)

$\sin (t^{2} h e a) \sin (t^{2} h e a)$ .

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Step 4.1.1.1

Raise

sin (t^{2} h e a)

$\sin (t^{2} h e a)$ to the power of

1

$1$ .

{sin}^{1} (t^{2} h e a) sin (t^{2} h e a) - - - 1

$\sin^{1} (t^{2} h e a) \sin (t^{2} h e a) - - - 1$

Step 4.1.1.2

Raise

sin (t^{2} h e a)

$\sin (t^{2} h e a)$ to the power of

1

$1$ .

{sin}^{1} (t^{2} h e a) {sin}^{1} (t^{2} h e a) - - - 1

$\sin^{1} (t^{2} h e a) \sin^{1} (t^{2} h e a) - - - 1$

Step 4.1.1.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

{sin (t^{2} h e a)}^{1 + 1} - - - 1

${\sin (t^{2} h e a)}^{1 + 1} - - - 1$

Step 4.1.1.4

Add

1

$1$ and

1

$1$ .

{sin}^{2} (t^{2} h e a) - - - 1

$\sin^{2} (t^{2} h e a) - - - 1$

{sin}^{2} (t^{2} h e a) - - - 1

$\sin^{2} (t^{2} h e a) - - - 1$

Step 4.1.2

Multiply

- - - 1

$- - - 1$ .

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Step 4.1.2.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

{sin}^{2} (t^{2} h e a) - 1 \cdot 1

$\sin^{2} (t^{2} h e a) - 1 \cdot 1$

Step 4.1.2.2

Multiply

- 1

$- 1$ by

1

$1$ .

{sin}^{2} (t^{2} h e a) - 1

$\sin^{2} (t^{2} h e a) - 1$

{sin}^{2} (t^{2} h e a) - 1

$\sin^{2} (t^{2} h e a) - 1$

{sin}^{2} (t^{2} h e a) - 1

$\sin^{2} (t^{2} h e a) - 1$

Step 4.2

Reorder

{sin}^{2} (t^{2} h e a)

$\sin^{2} (t^{2} h e a)$ and

- 1

$- 1$ .

- 1 + {sin}^{2} (t^{2} h e a)

$- 1 + \sin^{2} (t^{2} h e a)$

Step 4.3

Rewrite

- 1

$- 1$ as

- 1 (1)

$- 1 (1)$ .

- 1 (1) + {sin}^{2} (t^{2} h e a)

$- 1 (1) + \sin^{2} (t^{2} h e a)$

Step 4.4

Factor

- 1

$- 1$ out of

{sin}^{2} (t^{2} h e a)

$\sin^{2} (t^{2} h e a)$ .

- 1 (1) - 1 (- {sin}^{2} (t^{2} h e a))

$- 1 (1) - 1 (- \sin^{2} (t^{2} h e a))$

Step 4.5

Factor

- 1

$- 1$ out of

- 1 (1) - 1 (- {sin}^{2} (t^{2} h e a))

$- 1 (1) - 1 (- \sin^{2} (t^{2} h e a))$ .

- 1 (1 - {sin}^{2} (t^{2} h e a))

$- 1 (1 - \sin^{2} (t^{2} h e a))$

Step 4.6

Rewrite

- 1 (1 - {sin}^{2} (t^{2} h e a))

$- 1 (1 - \sin^{2} (t^{2} h e a))$ as

- (1 - {sin}^{2} (t^{2} h e a))

$- (1 - \sin^{2} (t^{2} h e a))$ .

- (1 - {sin}^{2} (t^{2} h e a))

$- (1 - \sin^{2} (t^{2} h e a))$

Step 4.7

Apply pythagorean identity.

- {cos}^{2} (t^{2} h e a)

$- \cos^{2} (t^{2} h e a)$

- {cos}^{2} (t^{2} h e a)

$- \cos^{2} (t^{2} h e a)$