Determine the coordinates of P for each of the following points.

Question

Point P(θ) is at the intersection of the unit circle and the terminal arm of angle θ

 

1.5π/6

2. P(5π/3)

3.P(-5Π/4)

Answer 1

The unit circle has a radius of 1.  We can use the formulas x =  r*cos(θ) and y =  r*sin(θ) to find the x and y coordinates.

 

1) x = 1*cos(5π/6) = -√3 / 2 and y = 1* sin(5π/6) = 1/2 → (-√3 / 2, 1/2)

 

I think you can get 2 and 3 by following what I have done in 1.

 

Let us start with some basic definitions. The unit circle is a circle whose center is at the origin with radius of 1. The angle θ is defined as the angle whose vertex is at the origin and whose initial side is the positive x axis. The terminal side is rotated either counterclockwise or clockwise from its initial side. Counterclockwise angles are positive and counter clockwise angles are negative. Consider a first quadrant angel θ. The terminal side intersects the unit circle at a point whose coordinates we will call (x,y). Drop a vertical line segment from the point to the positive x-axis forming a right triangle. The length of height of the triangle is y and the length of the base is x. Using soh cah toa we can write cos(θ) = x/1 which means cos(θ) = x and sin(θ) = y/1 which means sin(θ) = y. So the point (x,y) can be expressed as (cos(θ), sin(θ)). Also note that tan(θ) = y/x, x≠0. The trigonometric functions can be defined in terms of the unit circle. Why use the unit circle when we have soh cah toa? Note that soh cah toa only applies to angles that are between 0 to 90 degrees whereas the circle applies to all angles. Hope this helps. For further clarification I have include these tow websites.

https://www.mathsisfun.com/geometry/unit-circle.html

https://www.khanacademy.org/math/trigonometry/unit-circle-trig-func/unit-circle-definition-of-trig-functions/v/unit-circle-definition-of-trig-functions-1

Answer 2

Obtain a unit circle chart:

http://www.thattutorguy.com/unit-circle-pdf/