Download Chapter 7

Barnett/Ziegler/Byleen
College Algebra with Trigonometry, 6th Edition
Chapter Seven
Trigonometric Identities & Conditional Equations
Copyright © 1999 by the McGraw-Hill Companies, Inc.
Basic Trigonometric Identities
Reciprocal Identities
1
csc x = sin x
1
sec x = cos x
1
cot x = tan x
Quotient Identities
sin x
tan x = cos x
cos x
cot x = sin x
Identities for Negatives
sin(–x) = –sin x
cos(–x) = cos x
tan(–x) = –tan x
tan2 x + 1 = sec2 x
1 + cot2 x = csc2 x
Pythagorean Identities
sin2 x + cos2 x = 1
7-1-78
Suggested Steps in Verifying Identities
1. Start with the more complicated side of the identity,
and transform it into the simpler side.
2. Try algebraic operations such as multiplying, factoring,
combining fractions, splitting fractions, and so on.
3. If other steps fail, express each function in terms of sine and
cosine functions, and then perform appropriate algebraic
operations.
4. At each step, keep the other side of the identity in mind.
This often reveals what you should do in order to get there.
7-1-79
Sum, Difference, and Cofunction Identities
Sum Identities
sin(x + y) = sin x cos y + cos x sin y
cos(x + y) = cos x cos y – sin x sin y
tan x + tan y
tan(x + y) = 1 – tan tan
x
y
Difference Identities
sin(x – y) = sin x cos y – cos x sin y
cos(x – y) = cos x cos y + sin x sin y
tan x – tan y
tan(x – y) = 1 + tan tan
x
y
Cofunction Identities

Replace




2 with 90° if x is in degrees.

cos  2

–

x

= sin x

sin  2

–

x

= cos x

tan  2


– x = cot x

7-2-80
Double-Angle Identities
Double and Half-Angle
Identities
sin 2x = 2 sin x cos x
cos 2x = cos2 x – sin2 x = 1 – 2 sin2 x = 2 cos2 x – 1
tan 2x =
2 tan x
2 cot x
2
=
=
1 – tan2 x cot2 x – 1 cot x – tan x
Half-Angle Identities
x
sin 2 = ±
1 – cos x
2
x
cos 2 = ±
1 + cos x
2
x
tan 2 = ±
1 – cos x
sin x
1 – cos x
=
=
1 + cos x 1 + cos x
sin x
x
where the sign is determined by the quadrant in which 2 lies.
7-3-81
Product-Sum Identities
1
sin x cos y = 2 [sin (x + y ) + sin (x – y )]
1
cos x sin y = 2 [sin (x + y ) – sin (x – y )]
1
sin x sin y = 2 [cos(x – y ) – cos(x + y )]
1
cos x cos y = 2 [cos(x + y ) + cos(x – y )]
Sum-Product Identities
sin x + sin y = 2 sin
x+y
x–y
cos
2
2
sin x – sin y = 2 cos
x+y
x–y
sin
2
2
cos x + cos y = 2 cos
x+y
x–y
cos
2
2
cos x – cos y = –2 sin
x+y
x–y
sin
2
2
7-4-82
Trigonometric Equations
y
y = cos x
1
y = 0.5
–4 
–2 
2
4
x
–1
cos x = 0.5 has infinitely many solutions for –  < x < 
y
y = cos x
1
0.5
2
–1
x
cos x = 0.5 has two solutions for 0 < x < 2
7-5-83
Some Suggestions for Solving
Trigonometric Equations
1. Regard one particular trigonometric function as a variable,
and solve for it.
2. Consider using algebraic manipulation such as factoring.
3. Consider using identities.
4. After solving for a trigonometric function, solve for the variable
following the procedures discussed in the preceding section.
7-5-84