A. sin (3pi/2 – 3pi/4) can be expanded to:
A. cos 3pi/2 cos 3pi/4 + sin 3pi/2 cos 3pi/4
B. sin 3pi/2 cos 3pi/4 – cos 3pi/2 sin 3pi/4
C. cos 3pi/2 cos 3pi/4 – sin 3pi/2 sin 3pi/4
D. sin 3pi/2 cos 3pi/4 + cos 3pi/2 sin 3pi/4
B. Thick expression is obtained on subtracting 4x+6 from 7+2x?
The Correct Answer and Explanation is :
Let’s solve each part of your query one at a time.
Part A: Trigonometry Problem
You are asked to expand the expression (\sin \left(\frac{3\pi}{2} – \frac{3\pi}{4}\right)) using the angle subtraction formula for sine, which is:
[
\sin(a – b) = \sin a \cos b – \cos a \sin b
]
Plugging in the values (a = \frac{3\pi}{2}) and (b = \frac{3\pi}{4}), we have:
[
\sin \left(\frac{3\pi}{2} – \frac{3\pi}{4}\right) = \sin \frac{3\pi}{2} \cos \frac{3\pi}{4} – \cos \frac{3\pi}{2} \sin \frac{3\pi}{4}
]
To evaluate this, remember the values of sine and cosine at these specific angles:
- (\sin \frac{3\pi}{2} = -1)
- (\cos \frac{3\pi}{2} = 0)
- (\sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2})
- (\cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2})
Substituting these into the formula gives:
[
(-1)(-\frac{\sqrt{2}}{2}) – (0)(\frac{\sqrt{2}}{2}) = \frac{\sqrt{2}}{2}
]
This matches the expansion:
[
\sin \frac{3\pi}{2} \cos \frac{3\pi}{4} + \cos \frac{3\pi}{2} \sin \frac{3\pi}{4}
]
So, the correct option is D.
Part B: Algebra Problem
The expression for subtracting (4x + 6) from (7 + 2x) is calculated as follows:
[
(7 + 2x) – (4x + 6)
]
Distribute the negative sign through the second term:
[
7 + 2x – 4x – 6
]
Combine like terms:
[
1 – 2x
]
This represents the result of the subtraction: “1 – 2x.” This result is simply obtained by direct subtraction and combining like terms. The subtraction involves distributing the minus sign across (4x + 6) and then consolidating the x terms and the constant terms independently.
Generating the Image
Now let’s create an image that visually represents the algebraic expression subtraction process:
- Image Description: A chalkboard with the algebraic expression (7 + 2x – (4x + 6)) written in white chalk, showing each step of the subtraction: distribution of the negative sign, combining like terms, and resulting in (1 – 2x). Include visual arrows and annotations explaining each step.
I’ll generate this image now.
Here’s the image representing the step-by-step subtraction of the algebraic expression (7 + 2x – (4x + 6)). Each step is clearly annotated to show how the terms are combined to arrive at the final expression (1 – 2x). You can view the visual explanation on the chalkboard in the image above.
