Question

On the unit circle, an angle terminates at the coordinate (x,y). What is the relationship between the x and y components of the coordinate and the trig functions sin and cos?How can we use those relationships to predict the sign (+/-) for these trig functions when an angle terminates within a given quadrant?

Answer 1

On the unit circle, an angle terminating at the coordinates (x, y) corresponds to the trigonometric functions sine (sin) and cosine (cos).

The relationships between the signs of x, y, sin(θ), and cos(θ) based on the quadrant help predict the signs of trigonometric functions for a given angle.

Relationship between the x and y and trig function

On the unit circle, an angle terminating at the coordinates (x, y) corresponds to the trigonometric functions sine (sin) and cosine (cos).

For any angle θ measured counterclockwise from the positive x-axis to the terminal side of the angle, the coordinates (x, y) on the unit circle are x=cos(θ) and y=sin(θ).

Here's how these relationships help predict the sign (+/-) of the trigonometric functions in each quadrant:

First Quadrant (0 to 90 degrees or 0 to π/2 radians):

Both x and y are positive.

sin(θ) and cos(θ) are both positive.

Second Quadrant (90 to 180 degrees or π/2 to π radians):

x is negative, y is positive.

sin(θ) is positive, cos(θ) is negative.

Third Quadrant (180 to 270 degrees or π to 3π/2 radians):

Both x and y are negative.

sin(θ) and cos(θ) are both negative.

Fourth Quadrant (270 to 360 degrees or 3π/2 to 2π radians):

x is positive, y is negative.

sin(θ) is negative, cos(θ) is positive.

These relationships between the signs of x, y, sin(θ), and cos(θ) based on the quadrant help predict the signs of trigonometric functions for a given angle.

They are particularly useful when determining whether the functions are positive or negative, which aids in calculating values without directly using the unit circle or a calculator.

Answer 2

The unit circle is represented below:

As can be observed in the figure above, for an angle (t), the relation between the trigonometric functions and x and y components is:

x = cos(t)

y = sen(t)

x is positive in the first and in the fourth quadrant. Thus, cos (t) will be positive for angles in the first and fourth quadrant.

y is positive in the first and in the second quadrant. Thus, sin (t) will be positive for angles in the first and second quadrant.