Solved on Feb 05, 2024

Determine $\cos(-405^\circ)$ using the unit circle. Define cosine function from unit circle point $(x, y)$ .

STEP 1

Assumptions
1. The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the coordinate plane.
2. The cosine function relates an angle to the x-coordinate of a point on the unit circle.
3. The angle $t$ is measured in degrees from the positive x-axis, counterclockwise for positive angles and clockwise for negative angles.

STEP 2

To answer part (a), we need to recall the definition of the cosine function in terms of the unit circle. The cosine of an angle $t$ is the x-coordinate of the corresponding point $P(x, y)$ on the unit circle.
$\cos t = x$
Therefore, the correct answer is C. $\cos t = x$ .

STEP 3

For part (b), we need to determine the equivalent angle to $-405^\circ$ that lies within the range of $0^\circ$ to $360^\circ$ because angles that differ by full rotations (multiples of $360^\circ$ ) have the same cosine value.

STEP 4

To find the equivalent angle, we can add or subtract multiples of $360^\circ$ until we get an angle in the desired range. Since $-405^\circ$ is negative, we add $360^\circ$ to find the equivalent positive angle.
$-405^\circ + 360^\circ = -45^\circ$

STEP 5

The angle $-45^\circ$ is still negative, so we need to add another $360^\circ$ to get a positive angle.
$-45^\circ + 360^\circ = 315^\circ$

STEP 6

Now we have the equivalent angle $315^\circ$ which is in the range of $0^\circ$ to $360^\circ$ . Next, we need to find the cosine of $315^\circ$ .

STEP 7

The angle $315^\circ$ corresponds to a point on the unit circle in the fourth quadrant where both x and y coordinates are positive. Since $315^\circ$ is $45^\circ$ away from the positive x-axis, we can use the known cosine value for $45^\circ$ .

STEP 8

The cosine of $45^\circ$ is $\frac{\sqrt{2}}{2}$ , and since $315^\circ$ is in the fourth quadrant, the cosine will still be positive.
$\cos 315^\circ = \frac{\sqrt{2}}{2}$

STEP 9

Therefore, the value of $\cos(-405^\circ)$ is the same as the value of $\cos(315^\circ)$ , which is $\frac{\sqrt{2}}{2}$ .
The value of $\cos \left(-405^{\circ}\right)$ is $\frac{\sqrt{2}}{2}$ .

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Determine $\cos(-405^\circ)$ using the unit circle. Define cosine function from unit circle point $(x, y)$ .

STEP 1

STEP 2

To answer part (a), we need to recall the definition of the cosine function in terms of the unit circle. The cosine of an angle $t$ is the x-coordinate of the corresponding point $P(x, y)$ on the unit circle.
$\cos t = x$
Therefore, the correct answer is C. $\cos t = x$ .

STEP 3

For part (b), we need to determine the equivalent angle to $-405^\circ$ that lies within the range of $0^\circ$ to $360^\circ$ because angles that differ by full rotations (multiples of $360^\circ$ ) have the same cosine value.

STEP 4

To find the equivalent angle, we can add or subtract multiples of $360^\circ$ until we get an angle in the desired range. Since $-405^\circ$ is negative, we add $360^\circ$ to find the equivalent positive angle.
$-405^\circ + 360^\circ = -45^\circ$

STEP 5

The angle $-45^\circ$ is still negative, so we need to add another $360^\circ$ to get a positive angle.
$-45^\circ + 360^\circ = 315^\circ$

STEP 6

Now we have the equivalent angle $315^\circ$ which is in the range of $0^\circ$ to $360^\circ$ . Next, we need to find the cosine of $315^\circ$ .

STEP 7

The angle $315^\circ$ corresponds to a point on the unit circle in the fourth quadrant where both x and y coordinates are positive. Since $315^\circ$ is $45^\circ$ away from the positive x-axis, we can use the known cosine value for $45^\circ$ .

STEP 8

The cosine of $45^\circ$ is $\frac{\sqrt{2}}{2}$ , and since $315^\circ$ is in the fourth quadrant, the cosine will still be positive.
$\cos 315^\circ = \frac{\sqrt{2}}{2}$

STEP 9

Therefore, the value of $\cos(-405^\circ)$ is the same as the value of $\cos(315^\circ)$ , which is $\frac{\sqrt{2}}{2}$ .
The value of $\cos \left(-405^{\circ}\right)$ is $\frac{\sqrt{2}}{2}$ .

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Determine cos⁡(−405∘)\cos(-405^\circ)cos(−405∘) using the unit circle. Define cosine function from unit circle point (x,y)(x, y)(x,y).

STEP 1

STEP 2

STEP 3

For part (b), we need to determine the equivalent angle to −405∘-405^\circ−405∘ that lies within the range of 0∘0^\circ0∘ to 360∘360^\circ360∘ because angles that differ by full rotations (multiples of 360∘360^\circ360∘) have the same cosine value.

STEP 4

STEP 5

The angle −45∘-45^\circ−45∘ is still negative, so we need to add another 360∘360^\circ360∘ to get a positive angle.−45∘+360∘=315∘-45^\circ + 360^\circ = 315^\circ−45∘+360∘=315∘

STEP 6

Now we have the equivalent angle 315∘315^\circ315∘ which is in the range of 0∘0^\circ0∘ to 360∘360^\circ360∘. Next, we need to find the cosine of 315∘315^\circ315∘.

STEP 7

The angle 315∘315^\circ315∘ corresponds to a point on the unit circle in the fourth quadrant where both x and y coordinates are positive. Since 315∘315^\circ315∘ is 45∘45^\circ45∘ away from the positive x-axis, we can use the known cosine value for 45∘45^\circ45∘.

STEP 8

The cosine of 45∘45^\circ45∘ is 22\frac{\sqrt{2}}{2}22​​, and since 315∘315^\circ315∘ is in the fourth quadrant, the cosine will still be positive.cos⁡315∘=22\cos 315^\circ = \frac{\sqrt{2}}{2}cos315∘=22​​

STEP 9

Therefore, the value of cos⁡(−405∘)\cos(-405^\circ)cos(−405∘) is the same as the value of cos⁡(315∘)\cos(315^\circ)cos(315∘), which is 22\frac{\sqrt{2}}{2}22​​.The value of cos⁡(−405∘)\cos \left(-405^{\circ}\right)cos(−405∘) is 22\frac{\sqrt{2}}{2}22​​.

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Determine $\cos(-405^\circ)$ using the unit circle. Define cosine function from unit circle point $(x, y)$ .

For part (b), we need to determine the equivalent angle to $-405^\circ$ that lies within the range of $0^\circ$ to $360^\circ$ because angles that differ by full rotations (multiples of $360^\circ$ ) have the same cosine value.

The angle $-45^\circ$ is still negative, so we need to add another $360^\circ$ to get a positive angle.
$-45^\circ + 360^\circ = 315^\circ$

Now we have the equivalent angle $315^\circ$ which is in the range of $0^\circ$ to $360^\circ$ . Next, we need to find the cosine of $315^\circ$ .

The angle $315^\circ$ corresponds to a point on the unit circle in the fourth quadrant where both x and y coordinates are positive. Since $315^\circ$ is $45^\circ$ away from the positive x-axis, we can use the known cosine value for $45^\circ$ .

The cosine of $45^\circ$ is $\frac{\sqrt{2}}{2}$ , and since $315^\circ$ is in the fourth quadrant, the cosine will still be positive.
$\cos 315^\circ = \frac{\sqrt{2}}{2}$

Therefore, the value of $\cos(-405^\circ)$ is the same as the value of $\cos(315^\circ)$ , which is $\frac{\sqrt{2}}{2}$ .
The value of $\cos \left(-405^{\circ}\right)$ is $\frac{\sqrt{2}}{2}$ .