Fundamental Identities

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    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

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    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.

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    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 .

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    Example 3: Verify the identity

    Example 4: Verify the identity

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    11 Other sums of trigonometric functions

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    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

    20 See also

    21 Notes

    22 References

    23 External links

    [edit] Notation[edit] 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

    Radia

    ns

    Grads33

    grad

    66

    grad

    133

    grad

    166

    grad

    233

    grad

    266

    grad

    333

    grad

    366

    grad

    http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=1http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=2http://en.wikipedia.org/wiki/Greek_lettershttp://en.wikipedia.org/wiki/Alpha_(letter)http://en.wikipedia.org/wiki/Alpha_(letter)http://en.wikipedia.org/wiki/Beta_(letter)http://en.wikipedia.org/wiki/Gammahttp://en.wikipedia.org/wiki/Gammahttp://en.wikipedia.org/wiki/Thetahttp://en.wikipedia.org/wiki/Anglehttp://en.wikipedia.org/wiki/Degree_(angle)http://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Grad_(angle)http://en.wikipedia.org/wiki/Degree_(angle)http://en.wikipedia.org/wiki/Degree_(angle)http://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Grad_(angle)http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=1http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=2http://en.wikipedia.org/wiki/Greek_lettershttp://en.wikipedia.org/wiki/Alpha_(letter)http://en.wikipedia.org/wiki/Beta_(letter)http://en.wikipedia.org/wiki/Gammahttp://en.wikipedia.org/wiki/Thetahttp://en.wikipedia.org/wiki/Anglehttp://en.wikipedia.org/wiki/Degree_(angle)http://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Grad_(angle)http://en.wikipedia.org/wiki/Degree_(angle)http://en.wikipedia.org/wiki/Degree_(angle)http://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Grad_(angle)
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    Degre

    es45 90 135 180 225 270 315 360

    Radians

    Grads50

    grad

    100

    grad

    150

    grad

    200

    grad

    250

    grad

    300

    grad

    350

    grad

    400

    grad

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

    [edit] 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.

    [edit] 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

    This article uses the following notation for inverse trigonometric functions:

    http://en.wikipedia.org/wiki/Degree_(angle)http://en.wikipedia.org/wiki/Degree_(angle)http://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Grad_(angle)http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=3http://en.wikipedia.org/wiki/Sinehttp://en.wikipedia.org/wiki/Sinehttp://en.wikipedia.org/wiki/Cosinehttp://en.wikipedia.org/wiki/Cosinehttp://en.wikipedia.org/wiki/Tangent_functionhttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=4http://en.wikipedia.org/wiki/Inverse_trigonometric_functionshttp://en.wikipedia.org/wiki/Inverse_functionhttp://en.wikipedia.org/wiki/Inverse_functionhttp://en.wikipedia.org/wiki/Degree_(angle)http://en.wikipedia.org/wiki/Degree_(angle)http://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Grad_(angle)http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=3http://en.wikipedia.org/wiki/Sinehttp://en.wikipedia.org/wiki/Cosinehttp://en.wikipedia.org/wiki/Tangent_functionhttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=4http://en.wikipedia.org/wiki/Inverse_trigonometric_functionshttp://en.wikipedia.org/wiki/Inverse_function
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    Functi

    onsin cos tan sec csc cot

    Invers

    e

    arcsi

    n

    arcc

    os

    arcta

    n

    arcs

    ec

    arcc

    sc

    arcco

    t

    [edit] 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:

    [edit] Related identities

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