Trigonometry is based on the mathematical concept of the unit circle. This is a circle drawn on the x-y coordinate plane, with its center at (0,0), with a radius of 1. By setting the radius equal to 1, the trigonometric functions can be measured directly.If you envision a unit circle, any point on that circle establishes a right triangle. From a selected point on the circle, draw a vertical line directly to the x-axis. Then from that point on the x-axis, draw a horizontal line connecting to the origin. These two lines, the vertical and the horizontal, serve as the legs of a right triangle. The radius of the circle that connects the point on the circle to the center at the origin is the hypotenuse of the right triangle.

Trigonometric functions still apply to triangles and lengths other than 1, but setting the radius equal to 1 makes calculating the ratios more direct.
 The sine function is the ratio of the leg opposite a chosen angle to the hypotenuse of the right triangle. On the unit circle, the sine is a way of measuring the vertical distance from the x-axis to the designated point. This is another way of saying that it is the y-coordinate of the chosen point.The sine of an angle is commonly abbreviated as “sin.” The angle of measurement is often labeled θ{\displaystyle \theta }, by convention, so you say you are measuring sin⁡θ{\displaystyle \sin \theta } or sin(θ){\displaystyle sin(\theta )}.

For example, if you select an angle, called θ{\displaystyle \theta }, of 30 degrees at the center of the unit circle, this would mark a point on the circle with coordinates (32,12){\displaystyle ({\frac {\sqrt {3}}{2}},{\frac {1}{2}})}. You can then say that sin⁡θ=12{\displaystyle \sin \theta ={\frac {1}{2}}}., The cosine function is the ratio of the leg adjacent to the chosen angle divided by the hypotenuse of the right triangle. On the unit circle, the cosine is the length of the horizontal leg, which is also the x-axis coordinate of the point on the circle.The cosine of an angle is commonly abbreviated as “cos.” You say you are measuring cos⁡θ{\displaystyle \cos \theta } or cos⁡(θ){\displaystyle \cos(\theta )}.

For example, if you select an angle θ{\displaystyle \theta } of 30 degrees at the center of the unit circle, this would mark a point on the circle with coordinates (32,12){\displaystyle ({\frac {\sqrt {3}}{2}},{\frac {1}{2}})}. You can then say that cos⁡θ=32{\displaystyle \cos \theta ={\frac {\sqrt {3}}{2}}}., The third common trigonometric function is the tangent. The tangent is the ratio of the two legs of the right triangle to each other, without reference to they hypotenuse. Specifically, for a chosen angle of a right triangle, the tangent is found by dividing the length of the leg opposite to the chosen angle over the leg adjacent to the chosen angle. On the unit circle, the tangent is equal to the y-coordinate divided by the x-coordinate.The tangent function is often abbreviated as “tan.” For a selected angle θ{\displaystyle \theta }, you say you are measuring tan⁡θ{\displaystyle \tan \theta } or tan⁡(θ){\displaystyle \tan(\theta )}.

For the example of an angle θ{\displaystyle \theta } of 30 degrees at the center of the unit circle, recall that the coordinates are (32,12){\displaystyle ({\frac {\sqrt {3}}{2}},{\frac {1}{2}})}. You can find the tangent by dividing the sine (y-coordinate) by the cosine (x-coordinate) as follows:


tan⁡θ=1232=33=0.577{\displaystyle \tan \theta ={\frac {\frac {1}{2}}{\frac {\sqrt {3}}{2}}}={\frac {\sqrt {3}}{3}}=0.577}.Note that reporting the result in terms of a fraction with the square root, such as 33{\displaystyle {\frac {\sqrt {3}}{3}}} is generally considered more precise and more accurate than rounding to a decimal like 0.577. For practically purposes, a three-place decimal may be acceptable.
 Occasionally, you may need alternate ratios than the cosine, sine and tangent. These alternative functions are inverses of the first three. They are less commonly used in basic calculations. However, in more advanced trigonometric work, they become essential. These functions are:Secant. This is abbreviated as “sec” and is equal to 1cos{\displaystyle {\frac {1}{cos}}}.

Cosecant. The cosecant is abbreviated as “csc” and is equal to 1sin{\displaystyle {\frac {1}{sin}}}.

Cotangent. The cotangent is abbreviated as “cot” and is equal to 1tan{\displaystyle {\frac {1}{tan}}}.
 When trying to remember the ratios of the primary functions sin, cos and tan, many students use the memory tool “SOHCAHTOA.” When broken into its parts, it provides the ratios as follows:

SOH stands for the initials of sin, opposite, hypotenuse, and calls to mind the ratio:


sin=oppositehypotenuse{\displaystyle \sin ={\frac {\text{opposite}}{\text{hypotenuse}}}}



CAH stands for the initials of cos, adjacent, hypotenuse, as follows:


cos=adjacenthypotenuse{\displaystyle \cos ={\frac {\text{adjacent}}{\text{hypotenuse}}}}



TOA stands for the initials of tan, opposite, adjacent, and represents the ratio:


tan=oppositeadjacent{\displaystyle \tan ={\frac {\text{opposite}}{\text{adjacent}}}}
Understand the unit circle.
Learn the sine relationship.
Review the cosine function.
Understand the tangent function.
Review the other ratios.
Learn the mnemonic device SOHCAHTOA.
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