Trigonometric identities and formulas are essential for understanding angles and their relationships in mathematics. This resource provides detailed definitions of six trigonometric functions, including sine, cosine, and tangent, along with their reciprocal identities. It covers negative angle identities, double angle identities, and Pythagorean identities, making it a valuable tool for students studying trigonometry. Ideal for high school and college students preparing for exams, this guide also includes cofunction identities and half-angle identities. Each section is designed to enhance comprehension of trigonometric concepts and their applications in various mathematical problems.

Key Points

  • Defines six trigonometric functions including sine, cosine, and tangent.
  • Explains negative angle identities and their significance in trigonometry.
  • Covers double angle identities with examples for better understanding.
  • Includes Pythagorean identities that are fundamental for solving equations.
  • Details cofunction identities to illustrate relationships between angles.
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Trigonometric Identities
MVCC Learning Commons IT129
Six Trigonometric Functions
Right triangle definitions, where
0 < 𝜃𝜃< 𝜋𝜋/2
sin 𝜃𝜃=
opp
hyp
csc 𝜃𝜃=
hyp
opp
cos 𝜃𝜃=
adj
hyp
sec 𝜃𝜃=
hyp
adj
tan 𝜃𝜃=
opp
adj
cot 𝜃𝜃=
adj
opp
Circular function definitions, where 𝜃𝜃 is any angle.
𝑟𝑟=
𝑥𝑥
+ 𝑦𝑦
sin 𝜃𝜃=
𝑦𝑦
𝑟𝑟
csc 𝜃𝜃=
𝑟𝑟
𝑦𝑦
cos 𝜃𝜃=
𝑥𝑥
𝑟𝑟
sec 𝜃𝜃=
𝑟𝑟
𝑥𝑥
tan 𝜃𝜃=
𝑦𝑦
𝑥𝑥
cot 𝜃𝜃=
𝑥𝑥
𝑦𝑦
Negative Angle Identities
sin
(
𝜃𝜃
)
= sin 𝜃𝜃
cos
(
𝜃𝜃
)
= cos 𝜃𝜃
tan
(
𝜃𝜃
)
= tan 𝜃𝜃
csc
(
𝜃𝜃
)
= csc 𝜃𝜃
sec
(
𝜃𝜃
)
= sec 𝜃𝜃
cot
(
𝜃𝜃
)
= cot 𝜃𝜃
Reciprocal Identities
cos 𝜃𝜃=
1
sec 𝜃𝜃
tan 𝜃𝜃=
1
cot 𝜃𝜃
sec 𝜃𝜃=
1
cos 𝜃𝜃
cot 𝜃𝜃=
1
tan 𝜃𝜃
Tangent and Cotangent Identities
tan 𝜃𝜃=
sin 𝜃𝜃
cos 𝜃𝜃
cot 𝜃𝜃=
cos 𝜃𝜃
sin 𝜃𝜃
Double Angle Identities
sin 2𝜃𝜃= 2 sin 𝜃𝜃cos 𝜃𝜃
cos 2𝜃𝜃= cos
𝜃𝜃sin
𝜃𝜃= 2 cos
𝜃𝜃1 = 1 2 sin
𝜃𝜃
tan 2𝜃𝜃=
2 tan 𝜃𝜃
1 tan
𝜃𝜃
Pythagorean Identities
sin
𝜃𝜃+ cos
𝜃𝜃= 1 tan
𝜃𝜃+ 1 = sec
𝜃𝜃
cot
𝜃𝜃+ 1 = csc
𝜃𝜃
Cofunction Identities
sin 󰇡
𝜋𝜋
2
𝜃𝜃󰇢= cos 𝜃𝜃
cos 󰇡
𝜋𝜋
2
𝜃𝜃󰇢= sin 𝜃𝜃
csc 󰇡
𝜋𝜋
2
𝜃𝜃󰇢= sec 𝜃𝜃
tan 󰇡
𝜋𝜋
2
𝜃𝜃󰇢= cot 𝜃𝜃
sec 󰇡
𝜋𝜋
2
𝜃𝜃󰇢= csc 𝜃𝜃
cot 󰇡
𝜋𝜋
2
𝜃𝜃󰇢= tan 𝜃𝜃
Half Angle Identities
sin
𝜃𝜃
2
= ±
1 cos 𝜃𝜃
2
cos
𝜃𝜃
2
= ±
1 + cos 𝜃𝜃
2
tan
𝜃𝜃
2
= ±
1 cos 𝜃𝜃
1 + cos 𝜃𝜃
Sum and Difference Formulas
sin 𝐴𝐴+ sin 𝐵𝐵= 2 sin
𝐴𝐴+ 𝐵𝐵
2
cos
𝐴𝐴𝐵𝐵
2
sin 𝐴𝐴sin 𝐵𝐵= 2 cos
𝐴𝐴+ 𝐵𝐵
2
sin
𝐴𝐴𝐵𝐵
2
cos 𝐴𝐴+ cos 𝐵𝐵= 2 cos
𝐴𝐴+ 𝐵𝐵
2
cos
𝐴𝐴𝐵𝐵
2
cos 𝐴𝐴cos 𝐵𝐵= 2 sin
𝐴𝐴+ 𝐵𝐵
2
sin
𝐴𝐴𝐵𝐵
2
Addition and Subtraction Formulas
sin
(
𝐴𝐴± 𝐵𝐵
)
= sin 𝐴𝐴cos 𝐵𝐵± cos 𝐴𝐴sin 𝐵𝐵
cos
(
𝐴𝐴± 𝐵𝐵
)
= cos 𝐴𝐴cos 𝐵𝐵sin 𝐴𝐴sin 𝐵𝐵
tan
(
𝐴𝐴± 𝐵𝐵
)
=
tan 𝐴𝐴± tan 𝐵𝐵
1 tan 𝐴𝐴tan 𝐵𝐵
Product Formulas
sin 𝐴𝐴sin 𝐵𝐵=
1
2
[
cos
(
𝐴𝐴𝐵𝐵
)
cos(𝐴𝐴+ 𝐵𝐵)
]
cos 𝐴𝐴cos 𝐵𝐵=
1
2
[
cos
(
𝐴𝐴+ 𝐵𝐵
)
+ cos(𝐴𝐴𝐵𝐵)
]
sin 𝐴𝐴cos 𝐵𝐵=
1
2
[
sin
(
𝐴𝐴+ 𝐵𝐵
)
+ sin(𝐴𝐴𝐵𝐵)
]
cos 𝐴𝐴sin 𝐵𝐵=
1
2
[
sin
(
𝐴𝐴+ 𝐵𝐵
)
sin(𝐴𝐴𝐵𝐵)
]
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FAQs

What are the six trigonometric functions and their definitions?
The six trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent. Sine is defined as the ratio of the opposite side to the hypotenuse in a right triangle, while cosine is the ratio of the adjacent side to the hypotenuse. Tangent is the ratio of the opposite side to the adjacent side. The reciprocal functions are cosecant, secant, and cotangent, which are defined as the inverses of sine, cosine, and tangent, respectively. Understanding these functions is crucial for solving trigonometric equations and problems.

How do negative angle identities work in trigonometry?
Negative angle identities express trigonometric functions for negative angles in terms of their positive counterparts. For example, sin(-θ) = -sin(θ) indicates that the sine of a negative angle is the negative of the sine of the angle itself. Similarly, cos(-θ) = cos(θ) shows that cosine is an even function, meaning it remains unchanged when the angle is negated. These identities are essential for simplifying expressions and solving equations involving trigonometric functions.

What are double angle identities and how are they used?
Double angle identities relate the trigonometric functions of double angles to single angles. For instance, the sine double angle identity states that sin(2θ) = 2sin(θ)cos(θ), which is useful for simplifying expressions in calculus and algebra. The cosine double angle identity has three forms: cos(2θ) = cos²(θ) - sin²(θ), cos(2θ) = 2cos²(θ) - 1, and cos(2θ) = 1 - 2sin²(θ). These identities help in solving problems involving angles that are multiples of a given angle.

What are Pythagorean identities and their applications?
Pythagorean identities are fundamental relationships between the squares of the sine, cosine, and tangent functions. The primary identity states that sin²(θ) + cos²(θ) = 1, which is derived from the Pythagorean theorem. Other identities include tan²(θ) + 1 = sec²(θ) and cot²(θ) + 1 = csc²(θ). These identities are widely used in trigonometric proofs, simplifying expressions, and solving equations, making them essential for students studying trigonometry.

What are cofunction identities in trigonometry?
Cofunction identities express the relationship between the trigonometric functions of complementary angles. For example, sin(π/2 - θ) = cos(θ) indicates that the sine of an angle is equal to the cosine of its complement. Similarly, cos(π/2 - θ) = sin(θ) shows that cosine and sine are cofunctions. These identities are useful for solving problems involving angles and can simplify calculations in trigonometric equations.

How do half-angle identities work?
Half-angle identities provide formulas for the sine, cosine, and tangent of half an angle. For instance, sin(θ/2) = ±√(1 - cos(θ))/2 and cos(θ/2) = ±√(1 + cos(θ))/2. These identities are particularly useful in calculus and algebra when dealing with integrals and solving equations involving trigonometric functions. They allow for the simplification of expressions and help in finding exact values for trigonometric functions at specific angles.