Question
The angle 45degrees is shown below in standard position, together with a unit circle.
The image shows a unit circle and a right triangle in quadrant 1 with a 45-degree angle in standard position. The hypotenuse, which is also the terminal side of the angle, has a length of 1. The terminal side intersects the circle at (square root of 2 over 2, square root of 2 over 2).
Use the coordinates of the point of intersection of the terminal side and the circle to compute sec45degrees
.
The image shows a unit circle and a right triangle in quadrant 1 with a 45-degree angle in standard position. The hypotenuse, which is also the terminal side of the angle, has a length of 1. The terminal side intersects the circle at (square root of 2 over 2, square root of 2 over 2).
Use the coordinates of the point of intersection of the terminal side and the circle to compute sec45degrees
.
Answers
GPT-4o mini
To compute \(\sec(45^\circ)\), we first need to recall the relationship between secant, cosine, and the coordinates of the point where the terminal side of the angle intersects the unit circle.
The secant function is defined as the reciprocal of the cosine function:
\[
\sec(\theta) = \frac{1}{\cos(\theta)}
\]
In the case of \(45^\circ\) (or \(\frac{\pi}{4}\) radians), the coordinates of the point of intersection on the unit circle are given as \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\). Here, the x-coordinate represents \(\cos(45^\circ)\) and the y-coordinate represents \(\sin(45^\circ)\).
Since \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\), we can now find secant:
\[
\sec(45^\circ) = \frac{1}{\cos(45^\circ)} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}}
\]
To simplify \(\frac{2}{\sqrt{2}}\), we multiply the numerator and the denominator by \(\sqrt{2}\):
\[
\frac{2}{\sqrt{2}} = \frac{2 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{2\sqrt{2}}{2} = \sqrt{2}
\]
Thus, the value of \(\sec(45^\circ)\) is:
\[
\sec(45^\circ) = \sqrt{2}
\]
The secant function is defined as the reciprocal of the cosine function:
\[
\sec(\theta) = \frac{1}{\cos(\theta)}
\]
In the case of \(45^\circ\) (or \(\frac{\pi}{4}\) radians), the coordinates of the point of intersection on the unit circle are given as \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\). Here, the x-coordinate represents \(\cos(45^\circ)\) and the y-coordinate represents \(\sin(45^\circ)\).
Since \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\), we can now find secant:
\[
\sec(45^\circ) = \frac{1}{\cos(45^\circ)} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}}
\]
To simplify \(\frac{2}{\sqrt{2}}\), we multiply the numerator and the denominator by \(\sqrt{2}\):
\[
\frac{2}{\sqrt{2}} = \frac{2 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{2\sqrt{2}}{2} = \sqrt{2}
\]
Thus, the value of \(\sec(45^\circ)\) is:
\[
\sec(45^\circ) = \sqrt{2}
\]