Question

draw the terminal side of an angle in standard position that measures - 135°. hint: first plot a point at the origin, then plot a second point anywhere on the terminal side. a. name a point on the terminal side of the angle. ( , ) b. find the distance from the origin to that point. answer exactly. r = c. name another angle that is coterminal with the angle you have drawn.

Explanation:

Step1: Understand angle in standard - position

An angle of \(- 135^{\circ}\) in standard position is measured clock - wise from the positive x - axis. A \(135^{\circ}\) angle is \(45^{\circ}\) short of \(180^{\circ}\). So, the terminal side of the \(-135^{\circ}\) angle lies in the third quadrant and makes a \(45^{\circ}\) angle with the negative x - axis.

Step2: Find a point on the terminal side

For a \(45 - 45-90\) triangle in the third quadrant, if we consider the ratio of the sides \(x:y:r = 1:1:\sqrt{2}\) (where \(r\) is the distance from the origin to the point \((x,y)\)). A point on the terminal side can be \((-1,-1)\) since in the third quadrant both \(x\) and \(y\) are negative and for a \(45^{\circ}\) angle in this quadrant, \(|x| = |y|\).

Step3: Calculate the distance from the origin to the point

The distance \(r\) from the origin \((0,0)\) to the point \((x,y)\) is given by the formula \(r=\sqrt{x^{2}+y^{2}}\). If \(x=-1\) and \(y = - 1\), then \(r=\sqrt{(-1)^{2}+(-1)^{2}}=\sqrt{1 + 1}=\sqrt{2}\).

Step4: Find a coterminal angle

Coterminal angles are found by adding or subtracting \(360^{\circ}\) to the given angle. If we add \(360^{\circ}\) to \(-135^{\circ}\), we get \(-135^{\circ}+360^{\circ}=225^{\circ}\).

Answer:

a. \((-1,-1)\)
b. \(\sqrt{2}\)
c. \(225^{\circ}\)