Trigonometry, a branch of mathematics, studies relationships involving lengths and angles of triangles. It’s a subject that has immense applications in various scientific and engineering fields. A fundamental part of trigonometry is trigonometric identities, mathematical equations involving trigonometric functions that hold true for all possible values of the variable they contain.
What are Trigonometric Identities?
A trigonometric identity is an equation that relates trigonometric functions to each other. These identities are true for all values of the variables involved. They are based on the properties of the right triangle and the unit circle.
Trigonometric identities can be classified into different categories, including reciprocal identities, Pythagorean identities, ratio identities, opposite angle identities, complementary angles identities, sum and difference identities, double angle identities, half angle identities, product-sum identities, and periodic identities.
What is an Identity?
In mathematics, an identity is an equation that is invariably true, regardless of the specific value of the variable it contains. An identity can be trivially true, such as x = x, or it can be usefully true, like the Pythagorean Theorem’s a² + b² = c². In the context of trigonometry, these identities allow you to express a trigonometric expression in a different, but equivalent, form.
List of Trigonometric Identities
There are numerous trigonometric identities, but the ones most commonly used and seen are listed below.
Reciprocal Identities
Reciprocal identities express a trigonometric function as the reciprocal of another. The following are the basic reciprocal identities:
- sin θ = 1/cosec θ
- cos θ = 1/sec θ
- tan θ = 1/cot θ
Pythagorean Identities
These identities are derived from the Pythagorean theorem and form the basis of many trigonometric proofs. The most important Pythagorean identities are:
- sin² θ + cos² θ = 1
- 1 + tan² θ = sec² θ
- 1 + cot² θ = cosec² θ
Ratio Identities
These identities express one trigonometric function as a ratio of two others. The basic ratio identities are:
- tan θ = sin θ/cos θ
- cot θ = cos θ/sin θ
Opposite Angle Identities
Opposite angle identities define the value of a trigonometric function at a negative angle. These identities are:
- sin(-θ) = -sin θ
- cos(-θ) = cos θ
- tan(-θ) = -tan θ
Complementary Angles Identities
Complementary angles sum up to 90°. The trigonometric identities for complementary angles are:
- sin (90° – θ) = cos θ
- cos (90° – θ) = sin θ
- tan (90° – θ) = cot θ
Supplementary Angles Identities
Supplementary angles sum up to 180°. The trigonometric identities for supplementary angles are:
- sin (180° – θ) = sin θ
- cos (180° – θ) = -cos θ
- tan (180° – θ) = tan θ
Sum and Difference of Angles Identities
These identities express the sine, cosine, and tangent of the sum or difference of two angles. They are:
- sin (A + B) = sin A cos B + cos A sin B
- cos (A + B) = cos A cos B – sin A sin B
- tan (A + B) = (tan A + tan B) / (1 – tan A tan B)
Double Angle Identities
These identities express the sine, cosine, and tangent of twice an angle in terms of the sine, cosine, and tangent of the angle itself. The basic double angle identities are:
- sin 2θ = 2 sin θ cos θ
- cos 2θ = cos² θ – sin² θ
- tan 2θ = 2 tan θ / (1 – tan² θ)
Half Angle Identities
Half angle identities express the sine, cosine, and tangent of half an angle in terms of the sine, cosine, and tangent of the angle itself. The basic half angle identities are:
- sin (θ/2) = ±√[(1 – cos θ)/2]
- cos (θ/2) = ±√(1 + cos θ)/2
- tan (θ/2) = ±√[(1 – cos θ)(1 + cos θ)]
Product-Sum Identities
These identities convert the product of two trigonometric functions into a sum or difference. They include:
- sin A cos B = ½[sin(A + B) + sin(A – B)]
- cos A cos B = ½[cos(A + B) + cos(A – B)]
- sin A sin B = ½[cos(A – B) – cos(A + B)]
Trigonometric identities of Products
These identities express the product of two trigonometric functions in terms of the sum or difference of two other trigonometric functions. They include:
- sin A sin B = ½[cos(A – B) – cos(A + B)]
- cos A cos B = ½[cos(A + B) + cos(A – B)]
- sin A cos B = ½[sin(A + B) + sin(A – B)]
Periodicity of Trigonometric Function
A periodic function is a function that repeats its values in regular intervals or periods. The most important trigonometric functions, sine, cosine and tangent, are all periodic:
- sin(x + 2π) = sin(x)
- cos(x + 2π) = cos(x)
- tan(x + π) = tan(x)
Proof of the Trigonometric Identities
The proofs of these trigonometric identities are derived from basic geometric principles, algebraic manipulation, and the properties of the trigonometric functions. For example, the Pythagorean identities can be proved using the Pythagorean theorem, and the double angle identities can be proved using the sum and difference identities.
Relation between Angles and Sides of Triangle
In addition to the identities relating the trigonometric functions themselves, there are also identities that relate the sides and angles of a triangle. These include:
Sine Rule
The sine rule states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle. It is expressed as:
a/sinA = b/sinB = c/sinC
Cosine Rule
The cosine rule relates the lengths of the sides of a triangle to the cosine of one of its angles. It is expressed as:
c² = a² + b² – 2ab cosC
Tangent Rule
The tangent rule relates the difference and sum of the sides of a triangle to the tangent of half the difference and sum of the two opposite angles. It is expressed as:
(a – b)/(a + b) = tan[(A – B)/2] / tan[(A + B)/2]
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