sec 7pi/4 without a calculator
Find the exact answer
sec 7pi/4
= 1/cos(7π/4)
7π/4 is in quad IV, where the cosine is positive
7π/4 is π/4 away from the x-axis
so cos7π/4) = 1/√2
so sec 7π/4 = √2
To find the value of sec(7π/4) without a calculator, we can use the unit circle and basic trigonometric identities.
First, let's convert 7π/4 to degrees. Since π radians is equivalent to 180 degrees, we can multiply 7π/4 by 180/π to get the angle in degrees:
7π/4 * 180/π = (7*180) / 4 = 315 degrees.
Next, we determine the reference angle. The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. To find the reference angle for 315 degrees, we subtract 360 from 315 until it falls within 0 to 360 degrees:
315 - 360 = -45 degrees.
Since cosine is positive in the fourth quadrant and secant is the reciprocal of cosine, we will use the reference angle of 45 degrees in the first quadrant.
On the unit circle, in the first quadrant, the cosine value is equal to the x-coordinate, and the secant is the reciprocal of cosine. The point on the unit circle at 45 degrees is (√2/2, √2/2), which means that the x-coordinate (and the cosine) is √2/2.
Finally, we find the reciprocal of √2/2 to get the value of sec(7π/4):
sec(7π/4) = 1 / (√2/2) = 2 / √2 = √2
Therefore, sec(7π/4) is equal to √2.
To find the exact value of sec(7π/4) without using a calculator, we need to use the unit circle and the properties of the trigonometric functions.
We'll start by locating the angle 7π/4 on the unit circle.
First, note that 7π/4 is equivalent to (2π + π/4). Since one full revolution around the unit circle is equal to 2π radians, an angle of 2π is equivalent to going around the circle once. Therefore, we can subtract 2π from 7π/4 to bring it within one revolution of the circle.
7π/4 - 2π = π/4
So, the angle π/4 is equivalent to 7π/4 on the unit circle.
Next, we'll determine the value of sec(π/4). Recall that sec(θ) is the reciprocal of the cosine function, so we need to find the value of cos(π/4) first.
On the unit circle, the point corresponding to π/4 has coordinates (cos(π/4), sin(π/4)). Since π/4 is a 45-degree angle, it forms a 45-45-90 right triangle in the first quadrant.
Using the properties of a 45-45-90 triangle, we know that the cosine and sine of π/4 are equal: cos(π/4) = sin(π/4) = √2/2.
Therefore, sec(π/4) = 1 / cos(π/4) = 1 / (√2/2) = 2/√2 = √2.
Therefore, sec(7π/4) = sec(π/4) = √2.