Evaluate. Write your answer in simplified, rationalized form. Do not round.
sec 45°
$\sec 45^\circ = \frac{1}{\cos 45^\circ} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \boxed{\sqrt{2}}$.
We have $\sec{45^\circ} = \frac{1}{\cos 45^\circ} = \boxed{\sqrt{2}}$.
To evaluate sec(45°), we need to find the value of the secant function at an angle of 45 degrees.
The secant function is the reciprocal of the cosine function. So, to find the value of sec(45°), we need to first find the value of cos(45°).
Using the special right triangle, which is an isosceles right triangle with angles of 45°, the adjacent side and opposite side are both equal in length.
Let's assume the length of each side of the triangle is 1 unit. Using the Pythagorean theorem, we can find the length of the hypotenuse:
hypotenuse² = adjacent side² + opposite side²
hypotenuse² = 1² + 1²
hypotenuse² = 1 + 1
hypotenuse² = 2
Taking the square root of both sides gives us:
hypotenuse = √2
Now, we can find the value of cos(45°):
cos(45°) = adjacent side / hypotenuse
cos(45°) = 1 / √2
cos(45°) = √2 / 2
Since sec(θ) = 1 / cos(θ), we can find sec(45°) as follows:
sec(45°) = 1 / cos(45°)
sec(45°) = 1 / (√2 / 2)
sec(45°) = 2 / √2
To rationalize the denominator, we multiply the numerator and denominator by √2:
sec(45°) = (2 / √2) * (√2 / √2)
sec(45°) = (2√2) / 2
sec(45°) = √2
Therefore, sec(45°) simplifies to √2 or sqrt(2).
To evaluate the trigonometric function sec (45°), we need to first understand what secant represents.
Secant (sec) is the reciprocal of cosine, so it can be found using the formula:
sec(x) = 1 / cos(x)
Therefore, to evaluate sec (45°), we need to find the value of cos (45°) first.
Using the unit circle or a trigonometric table, we can find that cos (45°) is equal to √2 / 2.
Now, we can substitute the value of cos (45°) into the formula:
sec (45°) = 1 / (√2 / 2)
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator (√2 / 2):
sec (45°) = (1 / (√2 / 2)) * (√2 / 2)
Applying the multiplication, we get:
sec (45°) = (√2 / 2) / (2 / 2)
Simplifying further, we find:
sec (45°) = √2 / 2
Therefore, the simplified, rationalized form of sec (45°) is √2 / 2.