Trigonometry is based on measuring the sides of a triangle and his angles. The three most basic functions in trigonometry are the sine, cosine and the tangent. These functions take two sides of the triangle and calculate their ratio.
In trigonometry, the triangle is usually placed on a circle called the unit circle to measure its sides and angles. The unit circle can be in either radians or degrees. The unit circle has a radius of 1.
The x,y plane is imposed on the unit circle and is divided into quadrants. The quadrants are the four divisions of the unit circle and go counterclockwise from the x axis. The x and y axes mark the boundaries of quadrants.
The triangle is usually placed in the first quadrant although it can be in any other quadrant.
The triangles most frequently used in trigonometry are the right triangles.
The side opposite of the right angle is the hypotenuse. The length of the hypotenuse is equal to the radius of the unit circle. Because of this, the hypotenuse is sometimes symbolized by r. The right triangle also has an x side and a y side. The x side is horizontal along the x axis while the y side is vertical and parallel to the y axis. The x side is also referred to as adjacent while the y side is referred to as the opposite side due to the reference angle.
When the triangle is placed on the unit circle, the angle at the origin, which is the intersection of the x-y axes, becomes the reference angle.
The θ is the Greek letter Theta and it is used to represent the reference angle.
The sine function takes the ratio of the y side of the triangle to the hypotenuse.
| sinθ = |
y r |
The cosine function takes the ratio of the x side to the hypotenuse.
| cosθ = |
x r |
The tangent function takes the value of the y side and divides it by the x side.
| tanθ = |
y x |
There are additional trig functions which are reciprocal to the sine, cosine and tangent. These are the cosecant, secant and cotangent.
| secant : | secθ = |
1 cosθ |
= |
r x |
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| cosecant : | cscθ = |
1 sinθ |
= |
r y |
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| cotangent : | cotθ = |
1 cotθ |
= |
x y |