Module 4: Foundations of Trigonometry

 

In Try This 2 you found the following is true for a point P(x, y) on the unit circle:

 

 

 

This information can be simplified to the following:

 

 
cos θ = x sin θ = y

 

This is a diagram of the unit circle with point P at theta equal to cos theta, sin theta labelled in quadrant 1. From the point, a right triangle is drawn with the radius labelled 1.

Source: Pre-Calculus 12. Whitby, ON: McGraw-Hill Ryerson, 2011.

Reproduced with permission.

So, you can write any point that lies on the unit circle at angle θ as P(θ) = (x, y) or P(θ) = (cos θ, sin θ).

 

Now that the relationship between a point P(x, y) and cos θ, sin θ, and tan θ has been identified, you can use this information to solve problems. An example of how to use the coordinates of a point on the unit circle to determine trigonometric ratios for an unknown angle follows.

 

Example: Determining Trigonometric Ratios Using a Point on the Unit Circle

 

The point P lies on the unit circle at the intersection of a terminal arm of angle θ in standard position. Determine the values of sin θ, cos θ, and tan θ leaving the ratios as fractions.

 

Solution

 

Draw a diagram.

 

 
This is a diagram of a unit circle with the point the square root of 7 divided by 4, negative 3 quarters labeled on the circle.

 

Point P is on the unit circle.

 

Determine cos θ. Since point P is on the unit circle, cos θ = x. The x-coordinate is, therefore, equal to cos θ.


 

 

Determine sin θ. Since point P is on the unit circle, sin θ = y. The y-coordinate is, therefore, equal to sin θ.

 

 

 

Determine tan θ.

 

 

 


Self-Check 1

 

The point P  lies on the unit circle at the intersection of a terminal arm of angle θ in standard position.

  1. Sketch a diagram of point P on the unit circle, and label angle θ in standard position. Answers
  2. Determine the values of sin θ, cos θ, and tan θ. Leave the ratios as fractions. Answers