Lesson #1: Angles

By Rebecca Adcock


Two fundamental concepts in trigonometry are angle and angle measure. An angle is formed by rotating a ray around its endpoint. The endpoint of the ray becomes the vertex of the angle. The starting position of the ray is called the initial side of the angle. The ending position is called the terminal side. An angle in standard position has its initial side on the positive x-axis so the vertex is at the origin.


Angles in Standard and Non-standard Position






Positive and Negative Angles


The measure of an angle describes the magnitude and direction of the rotation of the ray from its initial position to its terminal position. If the rotation is counterclockwise, the angle has a positive measure. If the rotation is clockwise, the angle has a negative measure. An angle in standard position is said to lie in the quadrant where the terminal side resides.









Coterminal Angles


One way to measure an angle is in degrees. An angle created by one complete counterclockwise revolution measures 360°.  One generated by one complete clockwise rotation is -360°.  An angle may have a degree measure that is multiple of 360° or a fractional part of it.


Two or more angles in standard position can share the same terminal side and have different degree measures. These angles are called coterminal.




If θ is the degree measure of an angle, then all angles coterminal with this angle have a degree measure of  θ+360k where k is an integer. Using this formula for the example above where θ is 50.91°, -309.09° = 50.91°+(-1)360°. (Value of k is -1 and the angle was created on the first revolution of a ray in a negative direction.)





 Again using the formula  “θ+360k where k is an integer”, we can verify that an angle with measure -669.09° is coterminal to an angle with measure 50.91°. Where θ is 50.91°,

-669.09° = 50.91°+(-2)360°. (Value of k is -2 and the angle on the second revolution of a ray in a negative direction.)


The point you need to remember from all of this is that there are an infinite number of angles that are coterminal to any given angle. We can go around in circles forever in either a negative or positive direction creating coterminal angles. The measures of all the angles will differ by multiples of 360 ° times some integer.




Reference Angles


A useful tool for finding values of trigonometric functions of certain angles (you’ll learn how to do that later) is the reference angle. The reference angle is the acute angle formed by the terminal side of a given angle and the x-axis. A reference angle may be the same as the given angle.











Things to remember about reference angles:


Š      A reference angle is always ACUTE.


Š      A reference angle is always measured to the x-axis. Remember it this way: measuring a reference angle to the x-axis is like measuring your height from the top of your head to the ground. That makes sense! Measuring a reference angle to the y-axis is like measuring your height from the top of your head to a tree. That doesn’t make sense!













…..Don’t measure a reference angle to the y-axis……………...measure it to the x-axis….





Check your understanding. See Lesson #1 in Lesson Assessments.



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