Question

How do find the reference angle of $\sec \left( 280{}^\circ \right)$.

Answer 1

We will first understand the concept of the reference angle as reference angle is the smallest possible angle formed by the x-axis and the terminal line, going either clockwise or counter clockwise. We will first find the quadrant in which the given angle lies using the circular trigonometric function. Also, note that reference angle is always less than $90{}^\circ$.

Complete step-by-step solution:
We will first recall the concept of reference angle. Reference angle is the smallest possible angle formed by the x-axis and the terminal line (the line that determines the end of the angle), going either clockwise or counter clockwise. Also, the reference angle is smaller than $90{}^\circ$.
Now, we will recall the concept of the circular function of the trigonometric. Every angle is measured from the positive part of the x-axis and it is always measured in an anti-clockwise direction.
We will first find the quadrant in which the given angle lies. Then, we will use the appropriate formula to find the reference angle.
When the angle lies in the first quadrant, the reference angle will be the same as the actual angle.
When angle lies in the second quadrant, the reference angle will be the same as $180{}^\circ$ - actual angle.
When the angle lies in the third quadrant, the reference angle will be the same as the actual angle - $180{}^\circ$ .
When angle lies in the fourth quadrant reference angle will be equal to $360{}^\circ$ - actual angle.
Since, we have to find the reference angle of $\sec \left( 280{}^\circ \right)$.
Since, we know that $280{}^\circ$ lies in the fourth quadrant so to calculate the reference angle of $280{}^\circ$, we will use the formula: reference angle = $360{}^\circ$ - actual angle.
So, reference angle for $280{}^\circ$ = $360{}^\circ -280{}^\circ$
$\Rightarrow$ reference angle of $280{}^\circ$ = $80{}^\circ$.
So, the reference angle of $\sec \left( 280{}^\circ \right)$ is $80{}^\circ$and it is measured from the positive x-axis in clockwise direction.
So, we can also say that $280{}^\circ$ is equivalent to $-80{}^\circ$.
Now, we know that secant is positive in the fourth quadrant i.e. $\sec \left( -\theta \right)=\sec \theta$ , where $\theta$ lies in the fourth quadrant.
$\begin{aligned} & \Rightarrow \sec \left( 280{}^\circ \right)=\sec \left( -80{}^\circ \right) \\\ & \Rightarrow \sec \left( -80{}^\circ \right)=\sec \left( 80{}^\circ \right) \\\ \end{aligned}$
Hence, the reference angle of $\sec \left( 280{}^\circ \right)=\sec \left( 80{}^\circ \right)$.
This is our required solution.

Note: Students are required to note that the reference angle is always less than zero as it is measured either from positive or negative x-axis which give the smallest angle formed by the x-axis and the terminal angle. Also, note that cosine and secant of angle which lies in the fourth quadrant gives positive value for that angle.