Pre Activity: Trigonometric Functions in Regions
Remembering Unit Circle:
Click on the generate button see different angles in different quadrants.
Remembering the trigonometric functions in Unit Circle:
Use the slider 
to choose any angle θ. As you move the point around the unit circle, observe how the right triangle changes.
The radius of the circle is 1, so the hypotenuse of every triangle is always 1.
This is why the unit circle is so useful in trigonometry!
to choose any angle θ. As you move the point around the unit circle, observe how the right triangle changes.
The radius of the circle is 1, so the hypotenuse of every triangle is always 1.
This is why the unit circle is so useful in trigonometry!
- When you drop a vertical line from the point to the x-axis, its length shows sin(θ). This segment measures how far the point is up or down from the x-axis.
- When you draw a horizontal line from the point to the y-axis, its length shows cos(θ). This segment measures how far the point is left or right from the y-axis.
- The ratio of the sides gives tan(θ) and cot(θ). Notice how the green line extends this ratio beyond the triangle.
- In the same way, the extended purple and red segments represent cosec(θ), and sec(θ).
Questions:
1. Explain why sin(θ) and cos(θ) can never be greater than 1.
2. What does the animation made you see about the relationship between sin and cos?
Idea...
As you move the slider from 0° all the way to 360°, watch what happens in the other quadrants as the angle keeps growing and not an acute angle anymore.
- What do you notice about the values of sinθ and cosθ when θ is in the 1st quadrant?
- As θ moves into the 2nd, 3rd and 4th quadrants, do you see some of the same numerical values of sinθ and cosθ appearing again (possibly with different signs)?