LESSON 9 – REFERENCE ANGLES
Sa Lesson 9, matutunan natin ang tungkol sa reference angles o mga sangguniang anggulo. Ipapakita sa iyo ng araling ito kung paano hanapin ang mga values ng anim na trigonometric functions (sine, cosine, tangent, cosecant, secant, at cotangent) ng iba pang mga anggulo na may mga sukat na multiples ng 30o, 45o, at 60o. Malalaman mo na ang kanilang mga values ay katulad din ng natutunan natin sa nakaraang aralin.
PAG-ARALAN AT SURIIN NATIN
Pagmasdan ang Figure 1 sa ibaba:
Look at the terminal side of ∠A (Angle A). What is the acute angle formed by this terminal side with the horizontal axis?
If your answer is ∠B, you are right. ∠B is the acute angle formed by the terminal side of ∠A and the horizontal axis. ∠B is called the reference angle of ∠A.
Pagmasdan ang Figure 2 sa ibaba:
Figure 2
(Image from analyzemath.com)
Ang mga given angles A sa itaas ay may kulay itim, samantalang kulay pula naman ang mga reference angles.
Mapupuna natin ang mga reference angles ay mga acute angles (anggulo na may sukat na mataas sa zero degrees pero mas mababa sa 90 degrees.) Ang reference angle ay nabuo ng terminal side (pinakadulong gilid) ng given angle at ng horizontal axis ( or x-axis).
TANDAAN ANG MGA SUMUSUNOD
1. Lahat ng reference angle r ay acute angles na may sukat na mas mataas sa zero degree pero mas mababa sa 90 degrees.
2. Sa First Quadrant (0o < θ < 90o), ang reference angle r ng given angle θ ay ang mismong given angle θ.
Halimbawa, kung ang given angle θ = 30o, ang reference angle r nito ay ∠30o din.
3. Sa Second Quadrant (90o < θ < 180o), mahahanap ang reference angle r ng given angle θ sa pamamagitan ng pagkuha ng supplementary angle nito o paggamit ng formula nito: r = 180o - θ
Halimbawa, kung θ = 130o , its reference angle r is:
r = 180o - θ ==> r = 180o – 130o ==> r =50o.
3. Sa Third Quadrant (180o < θ < 270o), mahahanap ang reference angle r ng given angle θ sa pamamagitan ng formulang ito: r = θ - 180o
Halimbawa, kung θ = 250o , its reference angle r is:
r = θ - 180o ==> r = 250o – 180o ==> r =70o.
4. Sa Fourth Quadrant ( 270o < θ < 360o), mahahanap ang reference angle r ng given angle θ sa pamamagitan ng formulang ito: r = 360o - θ
Halimbawa, kung θ = 305o , its reference angle r is:
r = 360o - θ==> r = 360o – 305o ==> r =55o.
SUBUKIN NATIN ITO
Upang matasa ang ating natutunan, subukin nating sumagot ng ilang problema hinggil sa paghanap ng reference angle r.
1. What is the reference angle of 120o?
STEP 1: Determine in which interval the given angle belongs.
120o is between 90o and 180o. Kung gayon, ang 120o ay matatagpuan sa Second Quadrant o Quadrant II.
STEP 2: Determine which formula for r will be used.
Dahil ang 120o ay nasa Second Quadrant, ang formula nating gagamitin ay
r = 180o - θ
STEP 3: Substitute the value of the given angle.
ɵ = 120o
r = 180o - θ
r = 180o – 120o
r = 60o
Samakatuwid, ang reference angle r ng 120o ay ang special angle na 60o.
2. What is the reference angle of 330o?
STEP 1: Determine in which interval the given angle belongs.
300o is between 270o and 360o. Kung gayon ang 300o ay matatagpuan sa Fourth Quadrant o Quadrant IV.
STEP 2: Determine which formula for r will be used.
Dahil ang 330o ay nasa Fourth Quadrant, ang formula nating gagamitin ay
r = 360o - θ
STEP 3: Substitute the value of the given angle.
ɵ = 330o
r = 360o - θ
r = 360o – 330o
r = 30o
Samakatuwid, ang reference angle r ng 330o ay ang special angle na 30o.
3. What is the reference angle of 240o?
STEP 1: Determine in which interval the given angle belongs.
240o is between 180o and 270o. Kung gayon ang 240o ay matatagpuan sa Third Quadrant o Quadrant III.
STEP 2: Determine which formula for r will be used.
Dahil ang 240o ay nasa Third Quadrant, ang formula nating gagamitin ay
r = θ - 180o
STEP 3: Substitute the value of the given angle.
θ = 240o
r = θ - 180o
r = 240o – 180o
r = 60o
Samakatuwid, ang reference angle r ng 240o ay ang special angle na 60o.
MATUTO TAYO
A. The numeric value of the trigonometric function of a given angle is equal to the numeric value of the trigonometric function of its reference angle.
Ibig sabihin, kung ang reference angle r ng Ao ay Bo at ang sin Ao = x, ang sin Bo ay x din.
When we say numeric value, it means that we also consider the sign (positive or negative) of the trigonometric function.
B. We note that a trigonometric function is positive or negative depending on the measure of the angle. In determining the sign of a numeric value, we observe the following rules:
1. If the measure of θ is greater than 0° but less than 90°, all the six trigonometric functions (sin θ, cos θ, tan θ, csc θ, sec θ , and cot θ) are positive.
Kung ang given angle θ ay nasa First Quadrant, lahat ng anim na trigonometric functions - sin θ, cos θ , tanθ, csc θ, sec θ , at cot θ - ay pawang POSITIVE.
2. If the measure of θ is greater than 90° but less than 180°, sin θ and csc θ are positive. The other four functions are negative.
Kung ang given angle θ ay nasa Second Quadrant, ang sin θ at ang kanyang reciprocal/inverse na csc θ ay POSITIVE; ang iba pang function ay NEGATIVE na.
3. If the measure of θ is greater than 180° but less than 270°, tan θ and cot θ are positive. The other four functions are negative.
Kung ang given angle θ ay nasa Third Quadrant, ang tan θ at ang kanyang reciprocal / inverse na cot θ ay POSITIVE; ang iba pang function ay NEGATIVE na.
4. If the measure of θ is greater than 270° but less than 360°, cos θ and sec θ are positive. The other four functions are negative.
Kung ang given angle θ ay nasa Fourth Quadrant, ang cos θ at ang kanyang reciprocal / inverse na sec θ ay POSITIVE; ang iba pang function ay NEGATIVE na.
SUBUKIN NATIN ITO
Now let’s try to solve for the numeric values of the trigonometric functions of a given angle.
Problem 1: Find the values of the six trigonometric functions of the angle 120°.
We have previously learned that the reference angle of 120° is 60°.
Therefore, we will use the trigonometric function values of the special angle 60° that you learned in Lesson 8. Since 120° is in the Second Quadrant, only sine and its reciprocal/inverse cosecant are positive.
Thus, we have:
sin 120° = sin 60° = √3/2
csc 120° = csc 60° = 2/√3 𝑜𝑟 2√3/3
cos 120° = –cos 60° = - 1/2
sec 120° = –sec 60° = - 2/1 or -2
tan 120° = –tan 60° = - √3/1 𝑜𝑟 −√3
cot 120° = –cot 60° = - 1/√3 𝑜𝑟 −√3/3
Problem 2: Find the values of the six trigonometric functions of the angle 225°.
STEP 1: Determine the reference angle of the given angle.
225° is between 180° and 270° and in the Third Quadrant, so we use the formula:
r = θ - 180°
r = 225° - 180°
r = 45°
Samakatuwid, ang reference angle ng 225o ay ang special angle na 45o.
STEP 2: Determine the values of the trigonometric functions of the reference angle.
Gagamitin natin ang anim na trigonometric functions ng 45° na nakuha natin sa Lesson 8.
STEP 3: Determine the signs of the values.
The given angle is 225° and in the Third Quadrant. Therefore, tangent and cotangent are the only positive functions. We thus have:
sin 225° = -sin 45° = - 1/√2 or - √2/2
csc 225° = -csc 45° = - √2/1 𝑜𝑟 −√2
cos 225° = –cos 45° = - 1/√2 𝑜𝑟 −√2/2
sec 225° = –sec 45° = – √2/1 𝑜𝑟 −√2
tan 225° = tan 45° = 1/1 or 1
cot 225° = cot 45° = 1/1 or 1
Problem 3: Find the values of the six trigonometric functions of the angle 330°.
STEP 1: Determine the reference angle of the given angle.
330° is between 270° and 360° and in the Fourth Quadrant, so we use the formula:
r = 360o - θ
r = 360° - 330°
r = 30°
Samakatuwid, ang reference angle ng 330o ay ang special angle na 30o.
STEP 2: Determine the values of the trigonometric functions of the reference angle.
Gagamitin natin ang anim na trigonometric functions ng 30° na nakuha natin sa Lesson 8.
STEP 3: Determine the signs of the values.
The given angle is 330° and in the Fourth Quadrant. Therefore, cosine and secant are the only positive functions. We thus have:
sin 330° = -sin 30° = - 1/2
csc 330° = -csc 30° = - 2/1 𝑜𝑟 −2
cos 330° = cos 30° = √3/2
sec 330° = sec 30° = 2/√3 𝑜𝑟 (2√3)/3
tan 330° = -tan 30° = - 1/√3 or - √3/3
cot 330° = -cot 30° = - √3/1 or - √3
SUMMARY
Muling unawain at tandaan ang ating mga natutunan:
A. We use the following steps in computing for the reference angle r of a given angle θ:
STEP 1 Determine in which interval the given angle θ belongs or what quadrant it is located.
STEP 2 Determine which formula for r will be used.
STEP 3 Substitute the value of the given angle.
B. We use the corresponding formulas in computing for the reference angle according to the following rules.
1. If the given angle θ is greater than 90° but less than 180° (90° < θ < 180°), the formula to be used is:
r = 180°– θ .
2. If the angle θ is greater than 180° but less than 270° (180° < θ < 270°), the formula to be used is:
r = θ – 180°.
3. If the given angle θ is greater than 270° but less than 360° (270° < θ < 360°), the formula to be used is:
r = 360° – θ .
C. The numeric values of the trigonometric functions of a given angle is equal to the numeric values of the trigonometric functions of its reference angle.
In determining the signs of the numeric values of the trigonometric functions of a certain angle θ, we follow these rules:
1. If the measure of θ is greater than 90° but less than 180°, sin θ and csc θ are positive. The other four functions are negative. The given angle θ is in the Second Quadrant.
2. If the measure of θ is greater than 180° but less than 270°, tan θ and cot θ are positive. The other four functions are negative. The given angle θ is in the Third Quadrant.
3. If the measure of θ is greater than 270° but less than 360°, cos θ and sec θ are positive. The other four functions are negative. The given angle θ is in the Fourth Quadrant.
4. Upang madaling matandaan kung ano - ano lamang ang POSITIVE SIGNS ng numeric values ng mga Trigonometric Functions, tandaan lamang ang phrase na ito: “All Stations To Central” at ang drawing sa ibaba
PAGSASANAY
A. Find the reference angle of each of the following angles.
1. 135°
2. 150°
3. 210°
4. 300°
5. 315°
B. Determine the values of the following functions.
1. sin 135°
2. cos 210°
3. tan 240°
4. sec 315°
5. csc 330°
MGA SAGOT SA PAGSASANAY
A. Find the reference angle of each of the following angles.
1. 135°
135° is between 0o and 180o and it is in the 2nd Quadrant. The formula to be used is r = 180o – θ.
Since θ = 135o,
r = 180o – 135o
r = 45o
2. 150°
150° is between 0o and 180o and it is in the 2nd Quadrant. The formula to be used is r = 180o – θ.
Since θ = 150o,
r = 180o – 150o
r = 30o
3. 210°
210° is between 180o and 270o and it is in the 3rd Quadrant. The formula to be used is r = θ – 180o.
Since θ = 210o,
r = 210o – 180o
r = 30o
4. 300°
300° is between 270o and 360o and it is in the 4th Quadrant. The formula to be used is r = 360o – θ.
Since θ = 300o,
r = 360o – 300o
r = 60o
5. 315°
315° is between 270o and 360o and it is in the 4th Quadrant. The formula to be used is r = 360o – θ.
Since θ = 315o,
r = 360o – 315o
r = 45o
B. Determine the values of the following functions.
1. sin 135°
Step 1: Determine the reference angle of the given angle.
135° is between 0o and 180o and it is in the 2nd Quadrant. The formula to be used is r = 180o – θ.
Since θ = 135o, r = 180o – 135o = 45o. Therefore, the reference angle of 135o is the special angle 45o.
Step 2: Determine the values of the trigonometric functions of the reference angle.
sin 45o = 1/√2 𝑜𝑟 √2/2 [From Lesson 8]
Step 3: Determine the signs of the values.
The given angle is 135°. It is in the 2nd Quadrant, therefore, sine (and cotangent) are the only positive functions. We thus have:
sin 135o = sin 45o = 1/√2 𝑜𝑟 √2/2
We used the same steps above to find the values of the following:
2. cos 210°
r = θ – 180o = 210o – 180o = 30o
cos 30o = √3/2 [From Lesson 8]
210o is in the 3rd Quadrant, cosine is negative. Thus,
cos 210o = -cos 30o = - √3/2
3. tan 240°
r = θ – 180o = 240o – 180o = 60o
tan 60o = √3/1 𝑜𝑟 √3 [From Lesson 8]
240o is in the 3rd Quadrant, tan is positive. Thus,
tan 240o = tan 60o = √3/1 𝑜𝑟 √3
4. sec 315°
r = 360o – θ = 360o – 315o = 45o
sec 45o = √2/1 𝑜𝑟 √2 [From Lesson 8]
315o is in the 4th Quadrant, secant is positive. Thus,
sec 315o = sec 45o = √2/1 𝑜𝑟 √2
5. csc 330°
r = 360o – θ = 360o – 330o = 30o
csc 30o = 2/1 𝑜𝑟 2 [From Lesson 8]
330o is in the 4th Quadrant, cosecant is negative. Thus,
csc 330o = -csc 30o = - 2/1 𝑜𝑟 −2
SUSUNOD
Lesson 10 — Using a Scientific Calculator: Sin, Cos and Tan Keys/ Paggamit ng Scientific Calculator: Sin, Cos and Tan Keys
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