Question

How do you find the reference angle for 290 degrees?

Answer 1

Hint: The reference angle of an angle is the smallest angle that the terminal side of the given angle makes with X-axis. It should be noted that the reference angle is always a positive as well as acute means its measure always lies in the range \[\left[ {{0}^{\circ }},{{90}^{\circ }} \right]\]. Depending on the quadrant in which the original angle lies, the method to find the reference angle is different. Let \[a\] be the measure of the given angle. If the angle lies in the first quadrant then the reference angle equals \[a\], the angle itself. If the angle lies in the second quadrant then the reference angle equals \[{{180}^{\circ }}-a\]. If it lies in the third quadrant, then the reference angle equals \[a-{{180}^{\circ }}\]. If the angle lies in the fourth quadrant, then reference angle equals \[{{360}^{\circ }}-a\].

Complete step by step answer:
We are given the measure of an angle \[{{290}^{\circ }}\], we have to find its reference angle. First, we have to find in which quadrant does the angle lie, the measure of the angle is \[{{290}^{\circ }}\]. This value lies between the range \[\left[ {{270}^{\circ }},{{360}^{\circ }} \right]\], which is the range of the fourth quadrant. It means the angle lies in the fourth quadrant. We know that for an angle lying in the fourth quadrant, its reference angle equals \[{{360}^{\circ }}-a\], here \[a\] is the measure of the angle. Hence reference angle for the given angle will be calculated as,
\[\Rightarrow {{360}^{\circ }}-{{290}^{\circ }}={{70}^{\circ }}\]
Hence, the reference angle of \[{{290}^{\circ }}\] is \[{{70}^{\circ }}\].

Note:
 In trigonometry, we use the functions of angles like sin, cos, tan. It turns out that angles that have the same reference angles always have the same trig function values (the sign may vary). This comes in handy because we only then need to memorize the trig function values of the angles less than 90°. The rest we can find by first finding the reference angle.