Home > Do Mathematics -> Trigonometry -> The Cosine Function
Definition of Cosine
 Given a right-triangle (a triangle with a right angle) with acute angle α, the cosine of α is the value of the ratio: $\cos(\alpha)=\large{\frac{\text{side adjacent}\,\, \alpha}{\text{hypotenuse}}}=\large{\frac{\text{adj}}{\text{hyp}}}$

Definition of Cosine    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 cos(α) using your calculator. Use either the cos key and the size of the angle α or just divide the sides to find the value of the fraction $\frac{b}{c}$ (Both ways of calculating should give you the same value.)

Cosine - 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 adjacent to 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 adjacent to the angle α by the length of the hypotenuse. It should be the same number! This is the value of cos(α).

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

y=cos(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 x-coordinate of the point T ((width of the triangle) is the value of the cosine function of the angle α.
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