Click on the generate button see different angles in different quadrants.
Use the slider [icon]/images/ggb/toolbar/mode_slider.png[/icon] to choose any [b]angle θ.[/b] As you move the point around the unit circle, observe how the right triangle changes.[br][br]The radius of the circle is [b]1[/b], so the hypotenuse of every triangle is always [b]1[/b].[br][br]This is why the unit circle is so useful in trigonometry![br][list][*]When you drop a [b]vertical line[/b] from the point to the x-axis, its length shows [b]sin(θ)[/b].[br]This segment measures how far the point is [i]up or down[/i] from the x-axis.[br][br][/*][*]When you draw a [b]horizontal line[/b] from the point to the y-axis, its length shows [b]cos(θ)[/b].[br]This segment measures how far the point is [i]left or right[/i] from the y-axis.[br][br][/*][*]The ratio of the sides gives [b]tan(θ) and [b]cot(θ)[/b][/b].[br]Notice how the green line extends this ratio beyond the triangle.[br][br][/*][*]In the same way, the extended purple and red segments represent [b]cosec(θ)[/b], and [b]sec(θ)[/b].[br][br][/*][/list][b]As you explore, focus on how each function corresponds to a specific length on the diagram.[br][br]The goal is to recall where the basic trig functions come from and how they are visualized on the unit circle.[/b]
1. Explain why sin(θ) and cos(θ) can never be greater than 1.[br]
The radius of a unit circle is 1, so the opposite and adjacent sides cannot be longer than that.[br]Therefore sin and cos must stay between –1 and 1.
2. What does the animation made you see about the relationship between sin and cos?
They come from the same triangle and change together as the point moves.[br]When one increases, the other decreases.
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.[br][list][*]What do you notice about the values of sinθ and cosθ when θ is in the 1st quadrant?[/*][/list][list][*]As θ moves into the 2nd, 3rd and 4th quadrants, do you see some of the same [b]numerical values[/b] of sinθ and cosθ appearing again (possibly with different signs)?[br][br][/*][/list][b]And here is something to think about:[/b][br][br]From 0° to 90°, sinθ increases, but in other quadrants it cannot keep increasing because on the unit circle sine must stay between –1 and 1.[br][br]So if the angle keeps getting larger but the sine value cannot, [b]what should we expect when we see something like sin(150°)?[/b][br][br][b][size=150][i][color=#0000ff]In our main activities, we will explain how knowing the sine or cosine of one acute angle in the 1st quadrant can help you predict the sine or cosine of other angles around the circle![/color][/i][/size][/b][br]