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MATHEMATICS
Quarter 4 - Module 2
Trigonometric Ratios of Special
Angles

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Mathematics – Grade 9
Alternative Delivery Mode
Quarter 5 – Module 2: Trigonometric Ratios of Special Angles
First Edition, 2020

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Mathematics
Quarter 4 - Module 2
Trigonometric Ratios of
Special Angles
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LEARNING COMPETENCIES:

▪ Finds the trigonometric ratios of special angles.

(M9GE-IV b-c-1)

LEARNING OBJECTIVES:

K: Recall the definition of the six trigonometric ratios.

S: Find trigonometric ratios of special angles.

A: Show satisfaction in finding the trigonometric ratios of special angles.

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 I

MULTUPLE CHOICE. Write the letter of the correct answer.

1. Determine the exact value of sin 30° tan 45° + tan 30° sin 60°.
a. -1 b. 0 c. 1 d. 2
2. What is the exact value of cos 30° sin 45° + sin 30° tan 30°?

a. b.

c. d.

3. Find the difference of csc230° and cot245°.

a. 2 b. 3 c. 4 d. 5
4. Find the exact value of sin 30 cos30 sin 60 cos60.
a. √2 b. c. d. √3
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5. What is the exact value of 2sin30°cos30°.

a. b. c. d.

’s In

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What Makes Us Special?

Given the angles of the triangles below, find the values of the six trigonometric ratios.

1. Let a be the leg of a 45º-45º-90º Triangle.

45°
Sin 45° = __________ csc 45° =
a c = a √2 __________

Cos 45° = __________ sec 45° =

__________
45°
b= a
Tan 45° = __________ cot 45° =
__________

2. Let a be the shorter leg of a 30° - 60° - 90° Triangle

60° Sin 30° = __________ csc 30° = __________

a c = 2a
Cos 30° = __________ sec 30° = __________
30° Tan 30° = __________ cot 30° = __________
b = a √3

3. Let a be the shorter leg of a 30° - 60° - 90° Triangle

60° Sin 60° = __________ csc 60° = __________

a c = 2a
Cos 60° = __________ sec 60° = __________
30° Tan 60° = __________ cot 60° = __________
b = a √3

’s New

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Look at the table below and analyze.

The table above shows the corresponding values of trigonometric ratios from each special
angles( 𝜃°, 30°, 45°, 60°, and 90°).

is It

Finding Trigonometric Ratios of Special Angles

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 In trigonometry, 0°, 30°, 45°, 60° and 90° are called as special angles and they always
lie in the first quadrant. They are frequently seen in applications and we can use geometry to
determine the trigonometric ratios of these angles.

Examples: Determine the exact values of each of the following:

a) sin30°tan45° + tan30°sin60°
b) 2sin30°cos30°
c ) sin²45° + sec²60°

Solutions:
a.) sin30°tan45° + tan30°sin60°
• locate first their corresponding ratios from the table on trigonometric
ratios:
sin30° = ½
tan45° = 1

• solve:
sin30°tan45° = (½) (1) =

• therefore, sin30°tan45° + tan30°sin60° = 1

b.) 2sin30°cos30°
• locate first their corresponding ratios from the table on trigonometric
ratios:
sin30° = ½

• solve:

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 2sin30° = 2(1) = 2 = 1
2 2

√𝟑
• therefore, 2sin30°cos30° =
𝟐

c ) sin²45° + sec²60°

• locate first their corresponding ratios from the table on trigonometric

ratios:

sec60° = 2

• solve:

sin²45°

sec²60° = (2)2 = 4
sin²45° + sec²60° = 1 + 4 = 5

• therefore, sin²45° + sec²60° = 5

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 ’s More

Determine the six trigonometric ratios for angle D in the right triangle below.

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D 3 F

I Have Learned

Fill-in the chart below. Write your answers on your notebook.

3 things I learned

2 things that interest me

1 application of what I learned

I Can Do

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In your notebook, make your own problem using the trigonometric ratios and the four
fundamental operations (addition subtraction, multiplication and division). Show your
solutions. Your output will be rated by your teacher using the scale of 1 – 10.

Connect Me!

Read, analyze, solve and write the letter of the correct answer in each problem until you could
form a phrase. The choices are on the table below its corresponding letter.

Find the exact value of the following:

1.) sin30° + csc60°

2.) 5sec30°tan60°
3.) 3sin45° + 4cos45°
4.) sec45°csc45° - 1

5.) 1 -

𝑡𝑎𝑛 45°
6.)
(𝑡𝑎𝑛 30° + 𝑡𝑎𝑛 60°)

7.) tan2 60° - 2tan2 45° - cot2 30° + 2sin2 30°

8.) 4 (sin4 30° + cos460°) - 3 (cos245° - sin290°)

9.) 6 cos290° + 3 sin290° + 4 tan245°

10.) 4 cot245 - sec260 + sin260 + cos260

A C E I L

−
3 0

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 O P S U Y

7 10 1 2

Find the exact value of the variable/s. If your answer is not an integer, leave it in simplest
radical form.

1.)
6 x

60°
12

a. 2 b. 12
c. d.6√3

2.)
x y

30°
20√3

a. x =10 b. x =10, y = 30

c. x = 30 d. x =30, y = 10

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What I What’s In What’s Assessment Additional
Know More Activities
1. C sin D 1. S 1. D
2. A 2. P 2. D
cos = cos
3. B 3. E
45°= D=
4. D 4. C
tan 45°=
5. A 5 5. I
csc 45° = 6. A
tan D =
7. L
csc D = 5 4
sec 45°= 8. Y
cot sec D = 5 9. O
3
45°= 1 2. sin 10. U
cot D = 3

30° = cos 4
30° =
2
tan 30° =
3

csc 30° = 2

sec 30°=
cot
30°= 3. sin
60° =
cos 60° =
tan 60° =
csc

60° = sec
60°= 2 cot
60°=
3

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References
Argel A. and Angeles A.,2017. A Spiral Approach Mathematics 9; Educational Resources
Corporation.

Bryant, M.,et al.,2014. Mathematics Grade 9 Learner’s Module. DepEd.IM

Faylogna, F., et al.,2014.Chapter 8 Right Triangle Trigonometry ,Understanding Mathematics

Grdae 9.

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