Trigonometric IdentityA series of free High School Trigonometry Video Lessons from Brightstorm.
Trigonometric Identities Identities are equations true for any value of the variable. Since a right triangle drawn in the unit circle has a hypotenuse of length 1, we define the trigonometric identiesx=cos(theta) and y=sin(theta). In the same triangle, tan(theta)=x/y, so substituting we get tan(theta)=sin(theta)/cos(theta), the tangent identity. Another key trigonometric identity sin2(theta) + cos2(theta)=1 comes from using the unit circle and the Pythagorean Theorem.
Reciprocal Trigonometric Functions There are three reciprocal trigonometric functions, making a total of six including cosine, sine, and tangent. The reciprocal cosine function is secant: sec(theta)=1/cos(theta). The reciprocal sine function is cosecant, csc(theta)=1/sin(theta). The reciprocal tangent function is cotangent, expressed two ways: cot(theta)=1/tan(theta) or cot(theta)=cos(theta)/sin(theta).
Graphing Reciprocal Trigonometric Functions The three cofunction identities are useful because they can be used to convert, for example, sine into cosine or any trig function into its cofunction. When graphing reciprocal trigonometric functions, first find the values of the original trig function. Take the reciprocal of each value and plot the ordered pair in the coordinate plane.
Using Trigonometric Identities When simplifying problems that have reciprocal trig functions, start by substituting in the identities for each. If possible, write tangent in terms of sine and cosine. Use algebra to eliminate any complex fractions, factor, or cancel common terms. When using trigonometric identities, make one side of the equation look like the other or work on both sides of the equation to arrive at an identity (like 1=1).
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