     # Fundamental Identities

• View
220

0

Embed Size (px)

### Text of Fundamental Identities

• 8/3/2019 Fundamental Identities

1/30

Fundamental IdentitiesIf an equation contains one or more variables and is valid for all replacement values of thevariables for which both sides of the equation are defined, then the equation is known as anidentity. The equationx2 + 2x =x(x + 2), for example, is an identity because it is valid for all

replacement values ofx.

If an equation is valid only for certain replacement values of the variable, then it is called aconditional equation. The equation 3x + 4 = 25, for example, is a conditional equation becauseit is not valid for all replacement values ofx. An equation that is said to be an identity withoutstating any restrictions is, in reality, an identity only for those replacement values for which bothsides of the identity are defined. For example, the identity

is valid only for those values of for which both sides of the equation are defined.

The fundamental (basic) trigonometric identities can be divided into several groups. First are thereciprocal identities. These include

Next are the quotient identities. These include

• 8/3/2019 Fundamental Identities

2/30

Then there are the cofunction identities. These include

Next there are the identities for negatives. These include

Finally there are the Pythagorean identities. These include

The second identity is obtained by dividing the first by cos2 , and the third identity is obtainedby dividing the first by sin2 . The process of showing the validity of one identity based onpreviously known facts is called proving the identity. The validity of the foregoing identitiesfollows directly from the definitions of the basic trigonometric functions and can be used toverify other identities.

No standard method for solving identities exists, but there are some general rules or strategiesthat can be followed to help guide the process:

1. Try to simplify the more complicated side of the identity until it is identical to the secondside of the identity.

2. Try to transform both sides of an identity into an identical third expression.

3. Try to express both sides of the identity in terms of only sines and cosines; then try tomake both sides identical.

4. Try to apply the Pythagorean identities as much as possible.

5. Try to use factoring and combining of terms, multiplying one side of the identity by anexpression that is equal to 1, squaring both sides of the identity, and other algebraictechniques to manipulate equations.

• 8/3/2019 Fundamental Identities

3/30

Example 1: Use the basic trigonometric identities to determine the other five values of thetrigonometric functions given that

Example 2: Verify the identity cos + sin tan = sec .

• 8/3/2019 Fundamental Identities

4/30

Example 3: Verify the identity

Example 4: Verify the identity

• 8/3/2019 Fundamental Identities

5/30

11 Other sums of trigonometric functions

• 8/3/2019 Fundamental Identities

6/30

12 Certain linear fractional transformations

13 Inverse trigonometric functions

13.1 Compositions of trig and inverse trig functions

14 Relation to the complex exponential function

15 Infinite product formulae

16 Identities without variables

16.1 Computing

16.2 A useful mnemonic for certain values of sines and cosines

16.3 Miscellany

16.4 An identity of Euclid

17 Calculus

17.1 Implications

18 Exponential definitions 19 Miscellaneous

19.1 Dirichlet kernel

19.2 Weierstrass substitution

21 Notes

22 References

 Notation Angles

This article uses Greek letters such asalpha (),beta (), gamma(), and theta () to representangles. Several different units of angle measureare widely used, including degrees,radians, andgrads:

1 full circle = 360 degrees = 2 radians = 400 grads.

The following table shows the conversions for some common angles:

Degre

es30 60 120 150 210 240 300 330

ns

66

133

166

233

266

333

366

• 8/3/2019 Fundamental Identities

7/30

Degre

es45 90 135 180 225 270 315 360

100

150

200

250

300

350

400

Unless otherwise specified, all angles in this article are assumed to be in radians, though anglesending in a degree symbol () are in degrees.

 Trigonometric functions

The primary trigonometric functions are the sineand cosineof an angle. These are sometimesabbreviated sin() and cos(), respectively, where is the angle, but the parentheses around theangle are often omitted, e.g., sin and cos .

The tangent (tan) of an angle is theratio of the sine to the cosine:

Finally, the reciprocal functionssecant (sec), cosecant (csc), and cotangent (cot) are thereciprocals of the cosine, sine, and tangent:

These definitions are sometimes referred to as ratio identities.

 Inverse functions

Main article: Inverse trigonometric functions

The inverse trigonometric functions are partialinverse functions for the trigonometric functions.For example, the inverse function for the sine, known as the inverse sine (sin1) orarcsine(arcsin or asin), satisfies

and

• 8/3/2019 Fundamental Identities

8/30

Functi

onsin cos tan sec csc cot

Invers

e

arcsi

n

arcc

os

arcta

n

arcs

ec

arcc

sc

arcco

t

 Pythagorean identityThe basic relationship between the sine and the cosine is the Pythagorean trigonometric identity:

where cos2means (cos())2 and sin2means (sin())2.

This can be viewed as a version of the Pythagorean theorem, and follows from the equationx2 +y2 = 1 for the unit circle. This equation can be solved for either the sine or the cosine:

 Related identities

Dividing the Pythagorean identity through by either cos2or sin2yields two other identities:

Recommended ##### Chapter 5 â€“ Analytic Page | 89 Chapter 5 â€“ Analytic Trigonometry Section 1 Using Fundamental Identities
Documents ##### Equivalent Trigonometric Expressions Fundamental Trigonometric Identities For example, consider csc(أگ)
Documents ##### Section/Topic5.1 Fundamental Identities CC High School Functions Trigonometric Functions: Prove and apply trigonometric identities Objective Students will
Documents ##### Central Bucks School District / Homepage 5.1 â€” Using Fundamental Trigonometric Identities RFAEW: With
Documents ##### Using Fundamental Identities MATH 109 - Precalculus S. Rook
Documents ##### The Fundamental Lemma for Unitary Paris-Sud . Introduction The Fundamental Lemma is a set of combinatorial identities which have been formulated by Langlands-Shelstad
Documents ##### 10.3 The Six Circular Functions and Fundamental Identities
Documents ##### Trigonometry III Fundamental Trigonometric Identities. By Mr Porter
Documents ##### Jeff Bivin -- LZHS Using Fundamental Trig Identities Verifying Identities And Solving Trig Equations By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org
Documents ##### Chapter 5.1. 5.1 Using Fundamental Identities In this chapter, you will learn how to use the fundamental identities to do the following: Evaluate trigonometric
Documents Documents