Reciprocal  Identities, Quotient Identities and Pythagorean Identities

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High School Math based on the topics required for the Regents Exam conducted by NYSED. How to derive and use the Reciprocal, Quotient, and Pythagorean Identities?

Fundamental Trigonometric Identities
The reciprocal identities, quotient identities and the Pythagorean identities 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 identities x=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.

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).

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).

Fundamental Trigonometric Identities: Reciprocal, Quotient, and Pythagorean Identities How to use The Reciprocal, Quotient, and Pythagorean Identities Reciprocal, Quotient & Pythagorean Identities
A lesson on finding the first eight fundamental trigonometric identities and using them to simplify expressions.