The unit test blueprint lists topics currently taught in the unit along with both readiness and supporting standards. The
suggested percentage is included to give teachers an idea of how much of the topic should be tested in the unit with
respect to the other topics. The percentage was found by taking into account time spent on the topic in this unit.
Sample problems are written to meet the rigor of the TEKS.
Unit: Analytical Trig
TEKS
Topics tested
Trig Identities
P.2D,
P.5M
Subject: Pre-Calculus
Suggested
Sample Test Questions
Percent
40%
Ann and Sue both verified the same trig identity.
Ann’s Work
cos 
 sec   tan 
1  sin 
1
sin 


cos  cos 
1  sin  1  sin 


cos  1  sin 
1  sin 2 

cos  (1  sin  )

(sin 2   cos 2  )  sin 2 
cos  (1  sin)
cos 2 
cos  (1  sin  )
cos 
cos 

1  sin  1  sin 

Sue’s Work
cos 
 sec   tan 
1  sin 
cos  1  sin 


1  sin  1  sin 
cos  (1  sin  )

1  sin 2 
cos  (1  sin  )

(sin 2   cos 2  )  sin 2 
cos  (1  sin  )

cos 2 
1  sin 

cos 
1
sin 


cos  cos 
sec   tan   sec   tan 
Which student(s) verification was correct? (P.5M)
A
B
*C
D
Ann was correct.
Sue was correct.
Both Ann and Sue were correct.
Neither Ann nor Sue were correct.
Andy simplified sin( x)  tan( x) using trig identifies.
1  sec( x)
Step 1
Step 2
Step 3
Step 4
sin( x)  tan( x)
1  sec( x)
sin x  tan x
1  sec x
sin x
sin x 
cos x
1
1
cos x
sin x 

sin x 
cos x 
cos x 


cos x  1  1

cos x 

sin x cos x  sin x
cos x  1
The unit test blueprint lists topics currently taught in the unit along with both readiness and supporting standards. The
suggested percentage is included to give teachers an idea of how much of the topic should be tested in the unit with
respect to the other topics. The percentage was found by taking into account time spent on the topic in this unit.
Sample problems are written to meet the rigor of the TEKS.
Step 5
sin x  cos x  1
Step 6
cos x  1
sin x
He graphed y  sin( x)  tan( x) on his calculator to verify his
1  sec( x)
work.
Which is a true statement concerning Andy’s work. (P.2D,
P.5M)
A
Andy simplified the expression correctly since the
graph verifies his work.
B
Andy’s graph does not verify his work. He made a
mistake on step 1.
*C
Andy’s graph does not verify his work. He made a
mistake on step 2.
D
Andy’s graph does not verify his work. He made a
mistake on step 4.
Complete the reasoning for each step of the verification.
(P.5M)
Reason
1
sec x  tan x 
sec x  tan x
1
sec x  tan x
sec x  tan x 

sec x  tan x sec x  tan x
sec x  tan x
sec x  tan x 
sec 2 x  tan 2 x
sec x  tan x
sec x  tan x 
(tan 2 x  1)  tan 2 x
sec x  tan x
sec x  tan x 
1
The unit test blueprint lists topics currently taught in the unit along with both readiness and supporting standards. The
suggested percentage is included to give teachers an idea of how much of the topic should be tested in the unit with
respect to the other topics. The percentage was found by taking into account time spent on the topic in this unit.
Sample problems are written to meet the rigor of the TEKS.
sec x  tan x  sec x  tan x
Answer:
Reason
1
sec x  tan x
1
sec x  tan x
sec x  tan x 

sec x  tan x sec x  tan x
sec x  tan x 
sec x  tan x
sec 2 x  tan 2 x
sec x  tan x
sec x  tan x 
(tan 2 x  1)  tan 2 x
sec x  tan x
sec x  tan x 
1
sec x  tan x 
sec x  tan x  sec x  tan x
Rubric: 0
1
2
4
Solving Trig
Equations
P.2A,
P.2C,
P.2P,
P.4E,
P.5M,
P.5N
60%
Identity, multiplied by
1
= sec x  tan x
sec x  tan x
Multiply.
Pythagorean identity
Subtraction.
Divide.
- All reasons incorrect
– Only 1 or 2 reasons correct
– 3 or 4 reasons correct
– All reasons correct
The graph below represents the graph of each side of the
equation cos x  sin x 1 on the interval [0, 2 ) .
Which is not a possible solution to the equation
cos x  sin x 1 over the interval  4 ,0 ? (P.2P, P.5N)
A
B
C
*D
3
3

2



2
The unit test blueprint lists topics currently taught in the unit along with both readiness and supporting standards. The
suggested percentage is included to give teachers an idea of how much of the topic should be tested in the unit with
respect to the other topics. The percentage was found by taking into account time spent on the topic in this unit.
Sample problems are written to meet the rigor of the TEKS.
What are all of the solutions to
sin 2 x  cos x  1  0 ? (P.2A, P.2P,
P.5N)
A
x  2n , n 
*B x    2n , n 
C
x    2n or x 
D
no solution

 n
2
, n
Given the following:
12

 
, for
13
2
15

tan    , for    

8
2
Find cos     . (P.4E, P.5M)
sin  

220
221
140

B
221
140
*C
221
220
D
221
A

2
The equation tan x  cos x  2 is solved over the interval
[0, 2 )
sec x
Equation
Step 1
Step 2
Step 3
tan 2 x
 cos x  2
sec x
cos x  tan 2 x  cos x  2


cos x tan 2 x  1  2


cos x sec 2 x  2
Step 4
cos x  2
or
Step 5
cos x  2
or
Step 6
Step 7
sec2 x  2
sec2 x  2
sec x   2
 3 5 7
x , , ,
4 4 4 4
In which step was a mistake made? (P.2P, P.5M, P.5N)
The unit test blueprint lists topics currently taught in the unit along with both readiness and supporting standards. The
suggested percentage is included to give teachers an idea of how much of the topic should be tested in the unit with
respect to the other topics. The percentage was found by taking into account time spent on the topic in this unit.
Sample problems are written to meet the rigor of the TEKS.
A
B
C
D
Step 1
Step 3
Step 4
No mistake was made.
The horizontal distance in feet a ball travels after it is hit can be
modeled by the function
d
v0 2 sin  2 
32
,
v0
represents the initial
velocity and  represents the angle the ball is hit from the
horizontal in degrees.
If a tennis ball, hit with an initial velocity of 45 feet per second,
travels a horizontal distance of 40 feet, at approximately what
angles could the ball have been hit? (P.5N)
I
19.603
II
39.205
III 70.397
A
B
*C
D
I only
II only
I and III
II and III
sin 2x  cos x by making a table of
values for y  sin 2 x and y  cos x .
Michelle solved the equation

0
y  sin 2
0



6
3
2
3
3
2
2
0
2
3

3
2
5
6

3
2

0
The unit test blueprint lists topics currently taught in the unit along with both readiness and supporting standards. The
suggested percentage is included to give teachers an idea of how much of the topic should be tested in the unit with
respect to the other topics. The percentage was found by taking into account time spent on the topic in this unit.
Sample problems are written to meet the rigor of the TEKS.
y  cos 
1
3
2
1
2
0

1
2

3
2
What are the solutions to sin 2x  cos x on the interval
[0, 2 ) ? (P.2P, P.5N)
A
B
C
*D

  0, , 
2
  5
 , ,
6 2 6

3
  0, ,  ,
2
2
  5 3
 , , ,
6 2 6 2
1