Home > Do Mathematics -> Trigonometry -> The Sine Function
Definition of Sine
 Given a right-triangle (a triangle with a right angle) with acute angle α, the sine of α is the value of the ratio: $\sin(\alpha)=\large{\frac{\text{side opposite}\,\, \alpha}{\text{hypotenuse}}}=\large{\frac{\text{opp}}{\text{hyp}}}$

Definition of Sine    Directions and Links
 Link to GeoGebraTube:   Teacher   Student Click and drag point A or point B to change the triangle. (Angle ACB is always a right 90° angle so C depends on A and B). Find the value of sin(α) using your calculator. Use either the sin key and the size of the angle α or just divide the sides to find the value of the fraction $\frac{a}{c}$ (Both ways of calculating should give you the same value.)

Sine - Nested Triangles    Directions and Links
 Link to GeoGebraTube:   Teacher   Student Click and drag the angle α to any value between 0° and 90°. Then click and drag b and b' to make any 2 right-triangles you want with angle α. Look at first triangle: Divide the length of the side opposite the angle α by the length of the hypotenuse. Write this number down. Look at the second triangle and again divide the lenth of the side opposite the angle α by the length of the hypotenuse. It should be the same number! This is the value of sin(α).

y=sin(x) & Unit Circle - Degrees    Directions and Links
 Link to GeoGebraTube:   Teacher   Student Move the slider to α=45°. The sin(45°)≈0.707. Do you know the exact value of sin(45°)? Move the slider to α=135°. The sin(135°)≈0.707. Do you know the exact value of sin(135°)? Now, let α=–45°. Move the slider to the corresponding angle between 0° and 360°. The sin(–45°) = sin(315°) ≈ –0.707. Do you know the exact value of sin(–45°)? Does this agree with the fact that sin(–α) = –sin(α)?