QuestionCalculate $A = \sin^2 45^\circ - \tan 45^\circ + \sqrt{2} \cos 45^\circ$ and $B = (2-i)^2(4i-1)$ .

Studdy Solution

STEP 1

Assumptions1. We know the trigonometric values for45 degrees. . We understand the operations of addition, subtraction, multiplication, and square roots.
3. We understand the operations with complex numbers.

STEP 2

First, we need to simplify the expression A. We start by replacing the trigonometric values for45 degrees.
$A=\sin ^{2}45^{\circ}-\tan45^{\circ}+\sqrt{2} \cos45^{\circ}$

STEP 3

We know that $\sin45^{\circ} = \cos45^{\circ} = \frac{1}{\sqrt{2}}$ and $\tan45^{\circ} =1$ . So we substitute these values into the expression.
$A=\left(\frac{1}{\sqrt{2}}\right)^{2}-1+\sqrt{2} \cdot \frac{1}{\sqrt{2}}$

STEP 4

Now, we simplify the expression.
$A=\frac{1}{2}-1+1$

STEP 5

Further simplify the expression.
$A=\frac{1}{2}$

STEP 6

Now, we need to simplify the expression B. We start by expanding the expression.
$B=(2-i)^{2}(4 i-1)$

STEP 7

Expand the square in the expression.
$B=(4-4i+i^{2})(4i-1)$

STEP 8

We know that $i^{2} = -1$ . So we substitute this value into the expression.
$B=(4-4i-1)(4i-1)$

STEP 9

Further simplify the expression.
$B=(3-4i)(4i-)$

STEP 10

Now, we distribute the multiplication across the terms.
$B=12i-3-16i^{2}+4i$

STEP 11

Again, we know that $i^{} = -$ . So we substitute this value into the expression.
$B=i-3-16(-)+4i$

STEP 12

Further simplify the expression.
$B=16+16i-$

STEP 13

Finally, simplify the expression.
$B=13+16i$ So, the solutions are $A=\frac{}{2}$ and $B=13+16i$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms