Trigonometric Identities and Formulas
Basic Trigonometric Ratios
In a right triangle, with respect to an angle θ :
Opposite: The side across from the angle θ .
Adjacent: The side next to the angle θ .
Hypotenuse: The side opposite the right angle.
The three basic trigonometric ratios are defined using these sides:
Opposite
Sine (sin θ ): Hypotenuse
Adjacent
Cosine (cos θ ): Hypotenuse
Opposite
Tangent (tan θ ): Adjacent
Example: 3-4-5 Right Triangle
Consider a right triangle with sides 3, 4, and 5, where the side opposite angle θ is 4, the adjacent side is 3,
and the hypotenuse is 5.
sin θ = 45
cos θ = 35
tan θ = 43
Reciprocal Identities
These identities relate pairs of trigonometric functions:
Cosecant (csc θ ): sin1 θ
Secant (sec θ ): cos1 θ
1
Cotangent (cot θ ): tan θ
The reverse is also true:
sin θ = csc1 θ
cos θ = sec1 θ
tan θ = cot1 θ
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, Example: 3-4-5 Right Triangle (Continued)
Using the values from the previous example:
1 1 5
csc θ = sin θ
=4/5 = 4
1 1
sec θ = cos θ
= 3/5
= 53
1 1
cot θ = tan θ
= 4/3 = 34
Quotient Identities
These identities express tangent and cotangent in terms of sine and cosine:
sin θ
tan θ = cos θ
cot θ = cos θ
sin θ
Example: 3-4-5 Right Triangle (Continued)
Using the values from the previous examples:
sin θ 4/5 4
tan θ = cos θ
= 3/5
= 3
cos θ 3/5 3
cot θ = sin θ
= 4/5
= 4
Pythagorean Identities
These identities are derived from the Pythagorean theorem (a2 + b2 = c2 ).
The three fundamental Pythagorean identities are:
2
1. sin θ + cos2 θ = 1
2. 1 + cot2 θ = csc2 θ
3. 1 + tan2 θ = sec2 θ
2
These identities can be used to derive other forms. For example, from sin θ + cos2 θ = 1:
sin2 θ = 1 − cos2 θ
cos2 θ = 1 − sin2 θ
Even and Odd Identities
These identities describe how trigonometric functions behave when their input angle is negated.
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Basic Trigonometric Ratios
In a right triangle, with respect to an angle θ :
Opposite: The side across from the angle θ .
Adjacent: The side next to the angle θ .
Hypotenuse: The side opposite the right angle.
The three basic trigonometric ratios are defined using these sides:
Opposite
Sine (sin θ ): Hypotenuse
Adjacent
Cosine (cos θ ): Hypotenuse
Opposite
Tangent (tan θ ): Adjacent
Example: 3-4-5 Right Triangle
Consider a right triangle with sides 3, 4, and 5, where the side opposite angle θ is 4, the adjacent side is 3,
and the hypotenuse is 5.
sin θ = 45
cos θ = 35
tan θ = 43
Reciprocal Identities
These identities relate pairs of trigonometric functions:
Cosecant (csc θ ): sin1 θ
Secant (sec θ ): cos1 θ
1
Cotangent (cot θ ): tan θ
The reverse is also true:
sin θ = csc1 θ
cos θ = sec1 θ
tan θ = cot1 θ
1/6
, Example: 3-4-5 Right Triangle (Continued)
Using the values from the previous example:
1 1 5
csc θ = sin θ
=4/5 = 4
1 1
sec θ = cos θ
= 3/5
= 53
1 1
cot θ = tan θ
= 4/3 = 34
Quotient Identities
These identities express tangent and cotangent in terms of sine and cosine:
sin θ
tan θ = cos θ
cot θ = cos θ
sin θ
Example: 3-4-5 Right Triangle (Continued)
Using the values from the previous examples:
sin θ 4/5 4
tan θ = cos θ
= 3/5
= 3
cos θ 3/5 3
cot θ = sin θ
= 4/5
= 4
Pythagorean Identities
These identities are derived from the Pythagorean theorem (a2 + b2 = c2 ).
The three fundamental Pythagorean identities are:
2
1. sin θ + cos2 θ = 1
2. 1 + cot2 θ = csc2 θ
3. 1 + tan2 θ = sec2 θ
2
These identities can be used to derive other forms. For example, from sin θ + cos2 θ = 1:
sin2 θ = 1 − cos2 θ
cos2 θ = 1 − sin2 θ
Even and Odd Identities
These identities describe how trigonometric functions behave when their input angle is negated.
2/6
