Name
Date
7.1 – 7.3 Practice Test
For 1 – 5, simplify the trig expression
1. sin2  cot2  + sin2 
2. 1 - 2 csc2x + csc4x
3.
cos x
1  sin x

1  sin x
cos x
sin 2 x  cos2 x
4.
sin 2 x  sin x cos x
5. csc3 x – csc2 x – csc x + 1
For 6-10, verify the given trig expressions.
6. sin3 x + sin x cos2 x = sin x


7. sin   x   tan( x)  cot( x)  = -csc x
2

8. sec4  - tan4  = 1 + 2tan2 
9.
10.
cos x
1  sin x

1  sin x
cos x
1 sec(  x)
  csc x
sin(  x)  tan(  x)
11. Find the exact values of each of the following:
a. Sin 75
= sin( 30 + 45) = sin(30)cos(45) + sin(45)cos(30) = (1/2)(√2/2) + (√2/2)(√3/2)
= (√2 + √6)/4
b. Cos 75
= cos(30 + 45) = cos(30)cos(45) – sin(30)sin(45) = (√3/2)(√2/2) – (1/2)(√2/2)
= √6/4 - √2/4 = (√6 - √2)/4
c. Tan 75
= (√2 + √6)/4/((√6 - √2)/4) = (√2 + √6)/ (√6 - √2)
12. Given sin u = 12/13 and cos u is positive; cos v = -4/5 and tan v is positive, find
the following: using triangles; cos(u) = 5/13; sin(v) = -3/5
a. Sin (u + v)
= sin(u)cos(v) + sin(v)cos(u) = (12/13)(-4/5) + (-3/5)(5/13) = -48/65 – 15/65
= -63/65
b. Cos (u – v)
= cos(u)cos(v) + sin(u)sin(v) = (5/13)(-4/5) + (12/13)(-3/5) = -20/65 – 36/65
= -56/65
c. Tan (v – u); tan(v) = ¾; tan(u) = 12/5
= (3/4 – 12/5)/(1 + (3/4 * 12/5)) = (-33/20)/(56/20) = -33/56
13. Given the sin w is –5/13 and w is in quadrant 3, find each of the following:
d. sin(2w); using triangles, cos(w) = -12/13;
2sin(w)cos(w) = (-5/13)(-12/13) = 2(60/169) = 120/169
e. cos(2w)
= 1 – 2sin2(w) = 1 – 2(-5/13)2 = 1 – 2(25/169) = 1 – 50/169 = 119/169
f. tan(2w)
= (tanv – tanu)/(1 + tanvtanu) = (3/4 – 12/5)/(1 + ¾ x 12/5) = (-33/20)/(56/20)
= -33/56
Cofunction Identities


cos x  sin   x 
2



csc x  sec   x 
2



cot x  tan   x 
2



sin x  cos   x 
2



sec x  csc   x 
2



tan x  cot   x 
2

Period Identities
sin (x + 2) = sin x
cos (x + 2) = cos x
tan (x + ) = tan x
csc (x + 2) = csc x
sec (x + 2) = sec x
cot (x + ) = cot x
Odd/Even Identities
sin (-x) = - sin x
cos (-x) = cos x
tan (-x) = -tan x
csc (-x) = - csc x
sec (-x) = sec x
cot (-x) = - cot x
Pythagorean Identities
sin2 x + cos2 x = 1
tan2 x + 1 = sec2 x
1 + cot2 x = csc2 x
Double Angle Identities
sin 2x = 2 sin x cos x
tan 2 x 
2 tan x
1  tan 2 x
cos 2x = cos2 x – sin2 x
or
or
Sum and difference formulas
sin(x + y) = sin x cos y + cos x sin y
sin(x – y) = sin x cos y – cos x sin y
tan( x  y ) 
tan x  tan y
1  tan x tan y
2 cos2 x – 1
1 – 2 sin2 x
cos(x + y) = cos x cos y – sin x sin y
cos(x – y) = cos x cos y + sin x sin y
tan( x  y) 
tan x  tan y
1  tan x tan y
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