Grade 10 Mathematics Teachers’ Book

Adjust the Angle: Move the slider to select different values of \(\alpha\text{.}\) What happens to the values of \(\sin \alpha\text{,}\) \(\cos \alpha\text{,}\) and \(\tan \alpha\) as the angle increases?

Choose specific angles such as \(0^\circ\text{,}\) \(30^\circ\text{,}\) \(45^\circ\text{,}\) \(60^\circ\text{,}\) and \(90^\circ\text{.}\) Compare the displayed values with what you would expect from a calculator or trigonometric table. Do they match?

Observe the triangle in the bottom board. As \(\alpha\) increases, how do the side lengths \(a\) (opposite), \(b\) (adjacent), and \(c\) (hypotenuse) change?

Look at the sine values as the angle increases from \(0^\circ\) to \(90^\circ\text{.}\) Do they increase or decrease? What pattern do you notice?

Now observe the cosine values over the same interval. How does their pattern compare with the sine values?

As \(\alpha\) approaches \(90^\circ\text{,}\) what happens to the tangent value? Why does the display show a Math Error at \(90^\circ\text{?}\) What does this suggest about \(\tan 90^\circ\text{?}\)

Using the triangle, explain how the ratios \(\sin \alpha = \frac{\text{opposite}}{\text{hypotenuse}}\text{,}\) \(\cos \alpha = \frac{\text{adjacent}}{\text{hypotenuse}}\text{,}\) and \(\tan \alpha = \frac{\text{opposite}}{\text{adjacent}}\) are reflected in the changing side lengths.