Department of Education
Region V
Division of Camarines Sur
GOA SCIENCE HIGH SCHOOL
Tagongtong, Goa, Camarines Sur
Third Quarterly Exam in TRIGONOMETRY
Name: ________________________________Section: _______________ Score: __________
Direction: Choose the letter of the correct answer. No erasure please.
1
1. What is the value of sin 𝑥 = − 2 ? 210
a. -90
2.
b. 90
Find the angle that correspond to 𝑐𝑜𝑠 −1
√2
2
d. -270
c. 60
d. 90
c. 180
d. -135
c. 105
d. 115
√3
.
2
a. 30
b. 45
3. What is the value of x in tan 𝑥 = −√3 ?
a. -120
b. 60
4. What is 𝑐𝑜𝑠 −1
c. 210
√3
+ sin−1 20?
a. 45
b. 180
For item 5- 8. Evaluate the following trigonometric inverse.
5. 𝑐𝑜𝑡 (cos−1
a.
−√3
2
)
−√2
b. −√𝟑
2
−1
6. cos (sin 210)
a. 600
7. tan−1(cot 90)
a. 00
8. csc (sec −1 2)
a. 1
c. 1
d. 0
b. 1200
c. 180
d. 3600
b. 300
c. 450
d. 600
b.
−√2
c.
2
𝟐√𝟑
𝟐
d.
−√3
2
9. Which of the following shows a reciprocal identity relationship?
a.
𝟏
𝒄𝒐𝒔𝜽
= 𝒔𝒆𝒄𝜽
sin 𝜃
b. tan 𝜃 = cos 𝜃
c. cos −𝜃 = cos 𝜃
𝜋
d. cot 𝜃 = (tan − 𝜃)
2
10. Which of the following represents a correct Pythagorean Identity?
a. cos2 𝜃 − sin2 𝜃 = 1
b. cos2 𝜃 = sin2 𝜃 + 1
c. 𝐜𝐨𝐬𝟐 𝜽 − 𝟏 = 𝐬𝐢𝐧𝟐 𝜽
d. cos2 𝜃 / sin2 𝜃 = 1
11. Which of the following is a Product to Sum Identities of Sine function?
a. sin(𝜃 + 2𝜋𝑛) = sin(𝜃)
b. sin 𝛼 + sin 𝛽 = 2 sin(
𝛼+𝛽
2
) cos(
𝛼−𝛽
2
)
c. 𝐬𝐢𝐧(𝜶 + 𝜷) = 𝐬𝐢𝐧 𝜶 𝐜𝐨𝐬 𝜷 + 𝐜𝐨𝐬 𝜶 𝐬𝐢𝐧 𝜷
d.
1
2
[cos(𝛼 − 𝛽) − cos(𝛼 + 𝛽)] = sin 𝛼 sin 𝛽
12. Identify the cofunction formula of tangent function.
𝝅
a. 𝐭𝐚𝐧( 𝟐 − 𝜽) = 𝐜𝐨𝐭 𝜽
b. tan2 𝜃 = 1 − sec 2 𝜃
1− 𝐶𝑂𝑆 (2𝜃)
c. tan2 𝜃 =
1+𝐶𝑂𝑆 (2𝜃)
𝜃
1−cos 𝜃
d. tan( 2) = ±√1+cos 𝜃
13. Which is the Sum to product Identities of cosine function?
1
a. cos2 𝜃 = 2 (1 + cos 2𝜃)
b. 1cos 2𝜃 = cos 2 𝜃 − sin2 𝜃
c. cos(𝛼 ± 𝛽) = cos 𝛼 cos 𝛽 ∓ sin 𝛼 sin 𝛽
d. 𝐜𝐨𝐬 𝜶 − 𝐜𝐨𝐬 𝜷 = −𝟐 𝐬𝐢𝐧 (
𝜶+𝜷
𝜶−𝜷
𝟐
𝟐
) 𝐬𝐢𝐧(
)
14. What is the half – angle formula for Tangent function?
a. tan 𝛼 =
1
cot 𝜃
2 tan 𝛼
b. tan 2𝛼 = 1−tan2 𝛼
𝜽
𝟏−𝐜𝐨𝐬 𝜽
c. 𝐭𝐚𝐧(𝟐) = ±√𝟏+𝐜𝐨𝐬 𝜽
d. tan( 𝛼 + 𝛽) =
tan 𝛼+tan 𝛽
1−tan 𝛼 tan 𝛽
15. What is the solution of the equation −2 cos 𝑥 = √2 ? 135
a. 450
b. 2250
c. -1250
d. -450
For number 16 – 20. Give the simplified form of the following trigonometric expression.
16.
sec2 𝜃−1
sec2 𝜃
a. cos2 𝜃
17. cos2 𝜃 − 1
a. 𝐬𝐢𝐧𝟐 𝜽
b. 𝐬𝐢𝐧𝟐 𝜽
c. tan2 𝜃
d. 1
b. cos2 𝜃
c. tan2 𝜃
d. csc 2 𝜃
b. 0
c. undefined
d. 2
b. -1
c. − cos 2 𝜃
d. 1
b. cos 𝜃
c. cos2 𝜃
sin4 𝜃−cos4 𝜃
18. sin2 𝜃−cos2 𝜃
a. 1
sin2 𝜃+cos2 𝜃
19. csc2 𝜃−cot2 𝜃
a. cos2 𝜃
1
20. sec 𝜃 + cos 𝜃
a. 2
d. 2 𝐜𝐨𝐬 𝜽
cos2 𝜃
21. What identities/ formula must you first use to verifying the identity of 1+sin 0 ?
a. Even-Odd Identities
b. Reciprocal Identities
c. Half-Angle Identities
d. Pythagorean Identities
22. When will the equation sin2 𝜃 − 1 = 0 have a result of undefined?
a. If the constant term is less than 0.
b. If the constant term is less than 0 but greater than -1.
c. If the constant term is greater than 0 but less than 1.
d. If the constant term is greater than 1.
23. How many solutions does the equation 2 𝑠𝑖𝑛 𝜃 − 1 = 0
a. 1
b. 2
c. 3
d. 4
24. Solve the trigonometric equation 2 sin 𝜃 + √3 = 0
a. 3150
b. 3300
c. 4500
d. 3000
25. What is the solution of the trigonometric equation cos 2 𝜃 + sin 𝜃 = 1?
a. Undefined
d. 00
c. 900
`
2
2
26. What would you use to simplify the (sec 𝜃 − 1) cos 𝜃?
a. Reciprocal identity
b. Quotient identity c. Pythagorean identity d. Difference Identity
For Item 27 – 31. Organize the solution in order to verify 1 – tan2 x = 2 sec2 x – sec4 x. Choose the letter from
inside the box that corresponds to the steps indicated in each number.
𝑎. (2 − sec 2 𝑥)(sec 2 𝑥)
𝑏. [1 − (sec 2 𝑥 − 1)](sec 2 𝑥)
𝑐. 2 sec 2 𝑥 − sec 4 𝑥
𝑑. (1 − tan2 𝑥)(1 + tan2 𝑥)
𝑒. [1 − sec 2 𝑥 + 1](sec 2 𝑥)
27. _____d________
Factor difference of two square.
28. _____e______
Pythagorean Identities.
29. ______b_______
Distributive property.
30. _______a______
Simplify
31. _______c______
Distributive Property.
32. What approach would you use to solve the equation 2 sin 𝑥 cos 𝑥 + 2 sin 𝑥 + cos 𝑥 + 1 = 0?
a. Factoring
b. Using Identities
c. Extracting the roots
d. Combining like terms.
33. How would you solve the equation tan2 𝑥 − 3 tan 𝑥 = −2?
I.
Equate the factored term to zero.
II.
Group all terms so that they are in the left side.
III.
Find the values of x.
IV.
Find the factor of the equation.
a. IV, II, I, III
b. IV, II, I, III
c. II, IV, I, III
d. II, I, IV III
2
34. What identity/ies can you use to solve the equation tan 𝜃 + sec 𝜃 = 3
a. Pythagorean
b. Pythagorean, reciprocal identities
c. Quotient identities, reciprocal identities
d. Tangent and Cotangent identities, even –odd identities
35. What does it mean when you solve an equation with a range of 0 ≥ x ≥ 2π.
a. The solution of the equation is between 0 and 1800.
b. The solution of the equation is found from 00 to 1800.
c. The solution of the equation can only be found from 0 to 3600.
d. The solution of the equation can only be found in between 0 and 3600
36. The solution below is faulty. Which part of the solution is wrong or should not be there?
2 csc 3 𝑥 = 4 csc 𝑥
a. 𝟐 𝐜𝐬𝐜 𝒙 − 𝟒𝐜𝐬𝐜 𝒙 = 𝟎
𝒃. 𝐜𝐬𝐜 𝒙 (𝟐 𝐜𝐬𝐜 𝟐 𝒙 − 𝟒 𝐜𝐬𝐜 𝒙) = 𝟎
csc 𝑥 = 0
2 csc 2 𝑥 − 4 = 0
𝒄. 𝒙 = 𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅
2 csc 2 𝑥 = 4
4
csc 2 𝑥 =
2
√csc 2 𝑥 = √2
csc 𝑥 = √2
𝝅 𝟑𝝅
𝒅. 𝒙 = ,
𝟒 𝟒
𝟑
37. What is the most important thing to consider when using factoring in solving a trigonometric
equation.
a. The number of terms.
b. The similarity in terms.
c. The numerical coefficient.
d. The trigonometric expression used.
38. How would you describe the relationship between of tangent to sine and cosine?
a. Tangent is the reciprocal of cosine and sine.
b. Tangent is the ratio of sine and cosine.
c. The sum of sine and cosine is tangent.
d. The sum of the square of cosine and sine is the square of tangent.
39. What is the function of theta in a trigonometric equation?
a. It symbolizes an angle.
b. It pertains to the length of a triangle given an angle.
c. It refers to trigonometric ratio that is missing in an equation.
d. It shows what kind of solution is being ask by the problem.
40. Which part of the solution is the most important in solving the trigonometric equation?
a. Understanding the problem.
b. Identifying what you are looking for.
c. Finding what are the given of the problem.
d. Identifying what process, you are going to use.
Part II. Do what is ask.
1. Show that the simplifies form of cos4 𝑥 − sin4 𝑥 is cos 𝑥 − sin 𝑥. Write the identities or properties
you use to simplify the trigonometric identities.
Rubric for scoring:
5
4
3
2
1
0
Student shows Student shows Student shows Student shows Student shows No answer
complete
significant
some
slight
a
little
understanding
understanding understanding understanding understanding
and correctly and applies the of the order of of the order of of the order of
applies
the order of
operations, but operations.
operations.
order
of operations to
the solutions Parts of the Parts of the
operations
to simplify
are not correct problem may problem may
simplify
expressions
indicating that be copied, but be copied and
expressions.
with only a
parts of the little progress no progress is
Student shows few minor
process are not is made in made
in
all steps in the errors.
understood.
simplifying
simplifying
simplification
Student shows Student shows the
the
process.
90% all or
70% of the expressions.
expressions.
nearly all steps steps in the Student shows Student shows
in the
simplification 50% of the 30% of the
simplification process.
steps in the steps in the
process
simplification simplification
process.
process.
2. How would you verify the trigonometric identity sec 2 𝑥 (1 − cos 2 𝑥 𝑧) = tan2 𝑥?
5
4
3
2
1
0
Student shows Student shows Student shows Student shows Student shows No answer
complete
significant
some
slight
a
little
understanding
understanding understanding understanding understanding
and correctly and applies the of the order of of the order of of the order of
applies
the order of
operations, but operations.
operations.
order
of operations to
the solutions Parts of the Parts of the
operations
to verify the
are not correct problem may problem may
verify
the identities with indicating that be copied, but be copied, and
identities.
only a few
parts of the little progress no progress is
Student shows minor errors.
process are not is made in made
in
all steps in the Student shows understood.
verifying the verifying the
verification
90% - 70% all Student shows identities.
identities
process.
or nearly all
60% - 40% of Student shows Student shows
steps in the
the steps in the 30% - 20% of 10% of the
verification
verification
the steps in the steps in the
process.
process.
verification
verification
process
process