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### Course: Precalculus>Unit 2

Lesson 2: Trigonometric identities on the unit circle

# Tangent identities: periodicity

Sal solves a problem by considering the periodicity of the tangent function. Created by Sal Khan.

## Want to join the conversation?

• So this whole time, we've been describing the slopes of the rays of these points on the unit circle with tan(x)? So the slope of the ray through 60 degrees is rad3/2/.5?
• what exactly is periodicity?
• Periodicity is the quality of being periodic; that is, the tendency to recur at intervals.
• Since tan ø = opp/adj = y/x and this is a unit circle I get how the numerator in tan 0.46 = 1/2 can be 1 but how can the denominator be 2?
• Think of it this way.
The tangent is a ratio of 2 sides of the triangle ( opposite / adjacent ).
The 1 / 2 represents that ratio ,so, instead of representing actual lengths of the sides, it just indicates that the adjacent side is twice the length of the opposite side.
So, for instance, the adjacent side could be .6 long and the opposite side then would be .3.
• is the terminal ray is the tangent of the subtended angle?
• No, Sal's not talking about "tangent" as being a line (or ray), but about the trigonometric function (tan), which can be viewed as the slope of the terminal ray.

There is, however, a reason to why the tangent function is called tangent.
A ray, 𝑅, that emanates from the origin and forms the angle 𝜃 (0 < 𝜃 < 𝜋/2) with the 𝑥-axis intersects the unit circle in point 𝑇.
The straight line that touches the unit circle in 𝑇 is a tangent of the unit circle.
This line intersects the 𝑥-axis in a point 𝑋.
It is also perpendicular to 𝑅, which means that the length of the line segment between 𝑇 and 𝑋 will be tan 𝜃.
• Why is the tan of pi + 0.46 also a positive slope? If you start at the origin, isn't it then going downward since the arrow is going from right to left in a downward direction?
• Yes but as you cross the pi angle, you enter into the 3rd quadrant, where the tangent of an angle is always +ve, which you can understand by :
Recollecting that sine and cosine of an angle will always be of the same sign (-ve) in the third quadrant. or the first quadrant (+ve), or
Imagining it as such that the line whose angle with the positive x-axis is a reflex angle, in counterclockwise sense, will come back to the first and third quadrant, similar to the initial position, assuming you slowly turn the line about the origin in a counterclockwise fashion, hence making the tangent of the angle effectively same. Therefore
tan(pi+0.46)=tan(0.46)
• How do you do this without drawing a circle or without eyeballing it?
• By definition, an angle with the same slope is going to have the same tangent value. So, you don't have to visualize it on a unit circle in order to determine if two angles have the same tangent value.
• What is the differences in (2 pi + theta) and ( 2 pi - theta) on the unit circle? Thanks in advance.
• So they are both different angles assuming theta != 0. So they will just be two different points on the unit circle.
• my videos got stuck and wont let me finish or count the points for it. what should I do?
• You should go to the Help Center (a link is at the bottom of all KA screens).
• The tangent of angle pi/2+0.46 is 2/1 correct?
an answer would be great I feel like my entire understanding of this concept is balancing on the factuality of this statement.