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)?

 y=sin(x) & Unit Circle - Radians  Directions and Links
Link to GeoGebraTube:   Teacher   Student
  • First notice the unit circle has a radius of 1.
  • Click and drag the slider for α. Note the size of α in degrees and in radians. (Although we wrote the word radians, it should NOT be written. There is no unit with radians.)
  • The ANGLE α is on the x-axis. The ANGLE α is the LENGTH OF THE ARC in the unit circle for the angle α in degrees. Read this sentence until you understand it. It is critical.
  • The y-coordinate of the point T ((height of the triangle) is the value of the sine function of the angle α.
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