Precalculus II - Midterm 2

which is identically equal to sec²(x)?

tan²(x)+1

sin²(x)+cos²(x)=

1

tan²(x)+1=

sec²(x)

1+cot²(x)=

csc²(x)

confunction identities

sin(pi/2-x)=cos(x)

cos(pi/2-x)=sin(x)

tan(pi/2-x)=cot(x)

tan(x)=

sin(x)/cos(x)

cot(x)=

cos(x)/sin(x)

Even identities

cos(-x)=cosx

sec(-x)=sec(x)

odd identities

sin(-x)=-sin(x)

csc(-x)=-csc(x)

tan(-x)=-tan(x)

cot(-x)=-cot(x)

which is identically equal to cot(x)?

cos(x)/sin(x)

which is identically equal to 1?

sin²(x)+cos²(x)

which is identically equal to tan(x)?

sin(x)/cos(x)

which is identically equal to 1-cos²(x)?

sin²(x)

which is identically equal to 1+csc²(x)cos²(x)?

csc²(x)

which is identically equal to 1-sec(x)cos³(x)?

sin²(x)

which is identically equal to 1+sec²(x)sin²(x)?

sec²(x)

determine if it is odd, even, or neither: cos(x)

even

determine if it is odd, even, or neither: sin(x)cos(x)

odd

determine if it is odd, even, or neither: sin²(x)

even

determine if it is odd, even, or neither: sin(x)+cos(x)

neither

true or false: sin(2t)=2sin(t)

false

true or false: tan(x) + cos(x)/1+sin(x)

false

true or false: (1-cos²B)(1+cot²B)=1

true

true or false: (sin(x)-cos(x))²=1

false

find the positive value of a and b for: tan²t - sin²t = sin^(a)t/cos^(b)t

sin(t) = 4

cos(t) = 2

use identities to simplify as much as possible: sec²(x)-1

(tan(x))²

use identities to simplify as much as possible: sin(x)tan(x)/cos(x)

(tan(x))²

use identities to simplify as much as possible: sec(x)cos(x)

1

find solution on the interval [0,2pi): 2=-2sinx+1

x=7pi/6, 11pi/6

find solution on the interval [0,2pi): 3cot³(x)-1=0

x=pi/3, 2pi/3, 4pi/3, 5pi/3

find solution on the interval [0,2pi): 5=2-4cosx

x=arccos(-3/4), 2pi-arccos(-3/4)

find solution on the interval [0,2pi): 2-2sin²x=1+cosx

x=0, 2pi/3, 4pi/3

find solution on the interval [0,2pi): 3tan(x/2)+3=0

x=3pi/2

find solution on the interval [0,2pi): tan²2x-2tan2x+1=0

x=pi/4, 5pi/4, 9pi/4, 13pi/4

find all solutions on the interval [0,2pi): 2cosx-1=0

x=pi/3, 5pi/3

find all solutions on the interval [0,2pi): sec4x-2=0

x=pi/12, 5pi/12, 7pi/12, 11pi/12

find all solutions on the interval [0,2pi): 2cos3x=1

x=pi/9, 5pi/9, 7pi/9, 11pi/9, 13pi/9

find all solutions on the interval [0,2pi): (cosx)(2sinx+1)=0

x=pi/2, 3pi/2, 7pi/6, 11pi/6

find all solutions on the interval [0,2pi): 2cos²(x)-cos(x)-1=0

x=0, 2pi/3, 4pi/3

find all solutions on the interval [0,2pi): 4sinxcosx+2sinx-2cosx-1=0

x= pi/6, 5pi/6, 2pi/3, 4pi/3

sum of angle identities: sin(a+b)

sin(a)sin(b)+cos(a)sin(b)

sum of angle identities: cos(a+b)

cos(a)cos(b)-sin(a)sin(b)

sum of angle identities: tan(a+b)

tan(a)+tan(b) / 1-tan(a)tan(b)

use sum of angle identities: cos75degrees

sqrt(6)-sqrt(2) / 4

use sum of angle identities: sin(pi/12_

sqrt(6)-sqrt(2) / 4

use sum of angle identities: sin(x+pi)

-sin(x)

use sum of angle identities: cos(x+3pi/2)

-sin(x)

use addition or subtraction identity to find the values of A and B:

sin(x+pi/6) = Asin(x)+Bcos(x)

A=sqrt(3)/2

B=1/2

use addition or subtraction identity to find the value of A:

sin(x-pi/2) = Acosx

-1

use addition or subtraction identity to find the values of A and B:

cos(3pi/7)cos(2pi/21)+sin(3pi/7)sin(2pi/21) = cos(pi/A) = B

A = 3

B = ½

use addition or subtraction identity to find the values of the trig expression:

(x+pi/6) + sin(x-pi/3)

0

use addition or subtraction identity to find the value of the trig expression:

sin(pi/12)

sqrt(6)-sqrt(2) / 4

use addition or subtraction identity to find the value of the trig expression:

sin(5pi/12)

sqrt(6)+sqrt(2) / 4

use addition or subtraction identity to find the value of the trig expression:

cos(3pi/8)

sqrt(2-sqrt(2)) / 2

use addition or subtraction identity to find the value of the trig expression:

cos(pi/12)

sqrt(2)+sqrt(6) / 4

law of sines

sin(a)/A = sin(b)/B = sin (c)/C

a=6, <C=60degrees, <A=55degrees:

find <B and side length b

<B = 65 degrees

side length b = 6.64

c=6, <C=120degrees, <B=50degrees:

find <A and side length a

<A = 10 degrees

side length a = 1.2034

distance BC = 220 ft, <B=115degrees, <C=27degrees.

what is the distance across the river?

162.2 ft

b=8, <C=120degrees, <A=40degrees:

find <B, side length a, and side length c

<B = 20

side length a = 15.05

side length c = 20.25

<A=50degrees, a=105, b=43:

A) how many triangles fit this data

B) find <B, <C and side length c

A) 1

B) <B = 18.3 degrees, <C = 111.7 degrees, side length c = 127.35

determine the number of triangles possible:

a=27,b=23,<A=35degrees

1

determine the number of triangles possible:

a=13, b=15, <A=52degrees

2

determine the number of triangles possible:

a=10.8, b=23, <A=42degrees

0

law of cosines

c²=a²+b²-2abcos(Y)

use law of cosines to solve:

<A=60degrees, b=4, Y=4

Find c, <a and B

c=2sqrt(7)

<a=79.11degrees

B=40.89degrees

use law of cosines to solve:

a=624, b=670, c=180

find <Y, <A, <B

<Y = 15.5 degrees

<A = 83.1 degrees

<B = 98.6 degrees