Fundamental Trigonometric IdentitiesMATH 160, Precalculus

J. Robert Buchanan

Department of Mathematics

Fall 2011

J. Robert Buchanan Fundamental Trigonometric Identities

Objectives

In this lesson we will learn to:

recognize and write the fundamental trigonometricidentities,

use the fundamental trigonometric identities to evaluatetrigonometric functions, simplify trigonometric expressions,and to rewrite trigonometric expressions.

J. Robert Buchanan Fundamental Trigonometric Identities

Pythagorean Identities

sin2 u + cos2 u = 1

1 + tan2 u = sec2 u

1 + cot2 u = csc2 u

J. Robert Buchanan Fundamental Trigonometric Identities

Quotient and Reciprocal Identities

tan u =sin ucos u

sin u =1

csc u

cos u =1

sec u

tan u =1

cot u

cot u =cos usin u

csc u =1

sin u

sec u =1

cos u

cot u =1

tan u

J. Robert Buchanan Fundamental Trigonometric Identities

Cofunction Identities

sin(π

2− u

)

= cos u

cos(π

2− u

)

= sin u

tan(π

2− u

)

= cot u

csc(π

2− u

)

= sec u

sec(π

2− u

)

= csc u

cot(π

2− u

)

= tan u

J. Robert Buchanan Fundamental Trigonometric Identities

Even/Odd Identities

sin (−u) = − sin u

cos (−u) = cos u

tan (−u) = − tan u

csc (−u) = − csc u

sec (−u) = sec u

cot (−u) = − cot u

J. Robert Buchanan Fundamental Trigonometric Identities

Example (1 of 9)

If csc θ = −5 and cos θ < 0 find the values of all sixtrigonometric functions.

J. Robert Buchanan Fundamental Trigonometric Identities

Example (1 of 9)

If csc θ = −5 and cos θ < 0 find the values of all sixtrigonometric functions.

sin θ =

cos θ =

tan θ =

cot θ =

sec θ =

csc θ = −5

J. Robert Buchanan Fundamental Trigonometric Identities

Example (1 of 9)

If csc θ = −5 and cos θ < 0 find the values of all sixtrigonometric functions.

sin θ = − 15

cos θ = − 2√

65

tan θ =

√6

12cot θ = 2

√6

sec θ = − 5√

612

csc θ = −5

J. Robert Buchanan Fundamental Trigonometric Identities

Example (2 of 9)

Factor and simplify the following expression.

sin2 x csc2 x − sin2 x =

J. Robert Buchanan Fundamental Trigonometric Identities

Example (2 of 9)

Factor and simplify the following expression.

sin2 x csc2 x − sin2 x = sin2 x(csc2 x − 1)

= sin2 x(cot2 x)

= sin2 x(

cos2 x

sin2 x

)

= cos2 x

J. Robert Buchanan Fundamental Trigonometric Identities

Example (3 of 9)

Factor and simplify the following expression.

sec4 x − tan4 x =

J. Robert Buchanan Fundamental Trigonometric Identities

Example (3 of 9)

Factor and simplify the following expression.

sec4 x − tan4 x = (sec2 x + tan2 x)(sec2 x − tan2 x)

= (sec2 x + tan2 x)(1)

= sec2 x + tan2 x

J. Robert Buchanan Fundamental Trigonometric Identities

Example (4 of 9)

Factor and simplify the following expression.

cos4 x − 2 cos2 x + 1 =

J. Robert Buchanan Fundamental Trigonometric Identities

Example (4 of 9)

Factor and simplify the following expression.

cos4 x − 2 cos2 x + 1 = (cos2 x − 1)2

= (− sin2 x)

= sin4 x

J. Robert Buchanan Fundamental Trigonometric Identities

Example (5 of 9)

Carry out the multiplication and simplify the followingexpression.

(cot x + csc x)(cot x − csc x) =

J. Robert Buchanan Fundamental Trigonometric Identities

Example (5 of 9)

Carry out the multiplication and simplify the followingexpression.

(cot x + csc x)(cot x − csc x) = cot2 x − csc2 x

= −(csc2 x − cot2 x)

= −1

J. Robert Buchanan Fundamental Trigonometric Identities

Example (6 of 9)

Perform the subtraction and simplify the following expression.

1sec x + 1

− 1sec x − 1

=

J. Robert Buchanan Fundamental Trigonometric Identities

Example (6 of 9)

Perform the subtraction and simplify the following expression.

1sec x + 1

− 1sec x − 1

=sec x − 1

(sec x + 1)(sec x − 1)− sec x + 1

(sec x − 1)(sec x + 1)

=sec x − 1 − (sec x + 1)

sec2 x − 1

=−2

tan2 x= −2 cot2 x

J. Robert Buchanan Fundamental Trigonometric Identities

Example (7 of 9)

Rewrite the following expression so that it is not in fractionalform.

5sec x + tan x

=

J. Robert Buchanan Fundamental Trigonometric Identities

Example (7 of 9)

Rewrite the following expression so that it is not in fractionalform.

5sec x + tan x

=5

(sec x + tan x)

(sec x − tan x)

(sec x − tan x)

=5(sec x − tan x)

(sec x + tan x)(sec x − tan x)

=5(sec x − tan x)

sec2 x − tan2 x

=5(sec x − tan x)

1= 5 sec x − 5 tan x

J. Robert Buchanan Fundamental Trigonometric Identities

Example (8 of 9)

Substitute x = 2 cos θ with 0 < θ < π/2 in the expression√

64 − 16x2

and simplify.

J. Robert Buchanan Fundamental Trigonometric Identities

Example (8 of 9)

Substitute x = 2 cos θ with 0 < θ < π/2 in the expression√

64 − 16x2

and simplify.

√

64 − 16x2 =√

64 − 16(2 cos θ)2

=√

64 − 64 cos2 θ

=√

64(1 − cos2 θ)

=√

64 sin2 θ

= 8 sin θ

since sin θ > 0 when 0 < θ < π/2.

J. Robert Buchanan Fundamental Trigonometric Identities

Example (9 of 9)

Rewrite the following expression as a single logarithm andsimplify the result.

ln | tan x | + ln | csc x | =

J. Robert Buchanan Fundamental Trigonometric Identities

Example (9 of 9)

Rewrite the following expression as a single logarithm andsimplify the result.

ln | tan x | + ln | csc x | = ln | tan x csc x |

= ln∣

∣

∣

∣

sin xcos x

1sin x

∣

∣

∣

∣

= ln

∣

∣

∣

∣

1cos x

∣

∣

∣

∣

= ln | sec x |

J. Robert Buchanan Fundamental Trigonometric Identities

Homework

Read Section 5.1.

Exercises: 1, 5, 9, 13, . . . , 113, 117

J. Robert Buchanan Fundamental Trigonometric Identities