Question

What is the horizontal shift of the function y equals 3 sine left-parenthesis 4 x minus pi right-parenthesis?
(1 point)
Responses

π units to the right
π units to the right

start fraction pi over 2 end fraction units to the right
Image with alt text: start fraction pi over 2 end fraction units to the right

start fraction pi over 3 end fraction units to the right
Image with alt text: start fraction pi over 3 end fraction units to the right

start fraction pi over 4 end fraction units to the right

Which is the exact value of cosine left parenthesis negative 75 degrees right parenthesis?
(1 point)
Responses

start fraction start root 6 end root plus start root 2 end root over 4 end fraction
Image with alt text: start fraction start root 6 end root plus start root 2 end root over 4 end fraction

start fraction start root 6 end root minus start root 2 end root over 4 end fraction
Image with alt text: start fraction start root 6 end root minus start root 2 end root over 4 end fraction

start fraction negative start root 6 end root plus start root 2 end root over 4 end fraction
Image with alt text: start fraction negative start root 6 end root plus start root 2 end root over 4 end fraction

negativew start fraction start root 6 end root plus start root 2 end root over 4 end fraction

Answer 1

To address your two questions:

1. **Horizontal Shift of the Function \( y = 3 \sin(4x - \pi) \)**:
The general form for a sine function with a horizontal shift is \( y = a \sin(b(x - c)) + d \), where \( c \) represents the horizontal shift. Rearranging \( 4x - \pi \) gives \( 4(x - \frac{\pi}{4})\), indicating a shift of \( \frac{\pi}{4} \) to the right. Therefore, the horizontal shift is:

**Answer**: \( \frac{\pi}{4} \) units to the right.

2. **Exact Value of \( \cos(-75^\circ) \)**:
The cosine function is even, which means that \( \cos(-\theta) = \cos(\theta) \). Thus, \( \cos(-75^\circ) = \cos(75^\circ) \).

To find \( \cos(75^\circ) \), we can use the cosine of the sum identity:
\[
\cos(75^\circ) = \cos(45^\circ + 30^\circ) = \cos(45^\circ)\cos(30^\circ) - \sin(45^\circ)\sin(30^\circ
\]
Substituting known values:
\[
\cos(75^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right)
= \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}
\]
This simplifies to:
\[
\cos(75^\circ) = \frac{\sqrt{6} - \sqrt{2}}{4}
\]

Therefore, the exact value of \( \cos(-75^\circ) \) is:

**Answer**: \( \frac{\sqrt{6} - \sqrt{2}}{4} \).