the trigonometric identities: Pythagorean identity, Tangent identity

the reciprocal trigonometric functions and the reciprocal identities

how to graph reciprocal trigonometric functions

how to use trigonometric identities to simplify trigonometric expressions

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 identies x = cos θ and y = sin θ. In the same triangle, tan θ = x/y, so substituting we get tan θ = sin θ/cos θ, the tangent identity. Another key trigonometric identity sin^{2} θ + cos^{2} θ =1 comes from using the unit circle and the Pythagorean Theorem.

How to use the unit circle to derive the tangent identity and the Pythagorean identity.

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 θ = 1/cos θ. The reciprocal sine function is cosecant, csc θ = 1/sin θ. The reciprocal tangent function is cotangent, expressed two ways: cot θ = 1/tan θ or cot θ = cos θ/sin θ.

How to define the reciprocal trigonometric functions, the reciprocal identities, and the Pythagorean identities.

Derive the Pythagorean Identities from the unit circle
sin^{2} θ + cos^{2} θ =1
tan^{2} θ + 1 = sec^{2} θ
1 + cot^{2} θ = csc^{2} θ

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.

How to use the unit circle to derive identities that are useful in graphing the reciprocal trigonometric functions.

This video shows how to graph the reciprocal trignometric functions (y = cscx, y = secx and y = cotx) using the y = sinx, y = cosx and y = tanx functions.

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).

This video explains how to simplify to trigonometric expressions. One is a product of trigonometric functions and one is a quotient of trigonometric expressions.
sin^{2} x cot x csc x
(cos^{2} x − 1)/(cos^{2} x tan^{2} x)

This video provides an example of simplify trigonometric expression. It shows that if we recognize the identities it will help in the simplification.
(1 − cos^{2} x)/(1 + cot^{2} x)

This video provides an example of simplify trigonometric expression using trig identities.
(sin^{2} x − tan^{2} x)/(tan^{2} x sin ^{2} x)

This video provides an example of simplify trigonometric expression.
(1 − tan^{2} x)/(1 + tan^{2} x) + 1

This video provides an example of simplify trigonometric expression.
sin x/(1 + cos x) + (1 + cos x)/sin x

How to simplify a trigonometric expression by converting to sines and cosines and using algebra.
(sec x + 1)/(sin x + tan x)

Simplifying Trigonometric Expressions Using Identities
a) tan^{3} x csc^{3} x

b) sec x cos x − cos^{2} x
c) (csc^{2} x − 1) (sec^{2} x sin^{2} x)

d) (csc^{2} x − 1) / csc^{2} x
e) cot^{2} x / csc^{2} x

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