Trigonometric Identities.

Trigonometric

Identities

Trigonometric Identity

Equalities that involve trigonometric functions and are true for every single value of the occurring variables.

 Identities involving certain functions of one or more angles.

3 Groups or Relation

Reciprocal RelationQuotient RelationPythagorean Relation

Reciprocal Relation

The inverse trigonometric functions are partial inverse functions for the trigonometric functions.

tanand cottherefore, tanθ and cotθ are reciprocals of each other. The same thing can be said about sinθ and cscθ as well as cosθ and secθ.

Since the product of a number and its reciprocal equals 1, these relations may also be written as:

tanθcotθ=1

cosθsecθ=1

sinθcscθ=1

Quotient Relation

Simplifying, . But .

So by transivity;

Since cotθ is the reciprocal of tanθ the quotient can be derived to get

Pythagorean Relation

The basic relationship between the sine and the cosine is the Pythagorean trigonometric identity: where cos2 θ means (cos(θ))2 and sin2 θ 

means (sin(θ))2.This can be viewed as a version of

the Pythagorean theorem, and follows from the equation x2 + y2 = 1for the unit circle.

By Pythagorean Theorem, . Dividing both members by r² results to . Since and , then,  

cos²θ + sin²θ=1 

Dividing both members or by x² you get;

1 + tan²θ = sec²θ

dividing by y², you get;

cot²θ + 1 = csc²θ

Activity

A. Fill in the blanks to complete the table.

The Fundamental Trigonometric Identities and Their Alternate Forms

sinθcscθ = 1 1.

2.

tanθcotθ = 1 3.

4. 5.

6. 7.

sin²θ + cos²θ = 1 8. cos²θ = 1 - sin²θ

9. tan²θ = sec²θ - 1 sec²θ - tan²θ = 1

1 + cot²θ = csc²θ cot²θ = csc²θ - 1 10.

B. Use the fundamental identities to find the values of the other trigonometric functions.

1. tanθcotθ = ___________

2. csc²θ = ____________

3. = ___________

4. cosθ = ____________

5. sinθ = ___________

AssignmentWhat are the terminologies used in the graphs of trigonometric function? Define each.Reference: Trigonometry pages 141-142