Yaadannoo boqonnaa fi gaaffii shaakalaa

Boqonnaa 4 Herrega kutaa 10: Hojiiwwan Trigonometric

Wanti fuula kana irra jiru hundi boqonnaa 4 kitaaba Herrega kutaa 10 tiif: wanta boqonnaan haguugu, yaadannoo gabaabaa, gaaffii shaakalaa deebii isaanii waliin, fi viidiyoo barnootaa bakka jirutti.

6
Kaardii: 10
Gaaffii shaakalaa boqonnaa 4: 186–236
Fuula kitaabaa boqonnaa 4

Boqonnaan 4 maal haguuga

1.Radian Measure of angle
2.Basic Trigonometric Function
3.Trigonometric Identities & Equation
4.Application

Yaadannoo gabaabaa boqonnaa 4

Kaardiiwwan fuula Herrega kutaa 10 irratti boqonnaa kana jala galmaa'an. Tokko tokkoon bani.

Foomulaa Dheerina SarkiiFoormulaa

l = r \theta

l = r \theta

$l$: Dheerina aarkii $\text{m}$
$r$: Raadiyeesii geengoo $\text{m}$
$\theta$: Kofa gidduugaleessaa raadiyaaniin $\text{rad}$

Yoom akka itti fayyadamtu

Raadiyeesii fi kofa gidduugaleessaa beekamu irraa dheerina sarkii shallaguuf foomulaa kanatti fayyadami.

Of eeggadhu

Kofni gidduugaleessaa shallaggii dura gara raadiyaaniitti jijjiiramuu qaba. Kallattiin digirii fayyadamuun deebii dogoggoraa kenna.

Grade 10 Mathematics, fuula 187-237 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Dogoggora Digirii fi Raadiyaanii Shallaggii Sarkii KeessattiDogoggora baratame

Yoom akka itti fayyadamtu

Yeroo gaaffilee dheerina sarkii fi sochii geengoo hojjettu dogoggora kana of eeggadhu.

Of eeggadhu

Foomulaa l = r theta keessatti kofa digiriin jiru osoo pi / 180 hin baay'isin kallattiin hin galchin.

Grade 10 Mathematics, fuula 187-237 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Foomulaa Double Angle SineFoormulaa

\sin 2\theta = 2\sin\theta\cos\theta

\sin 2\theta = 2\sin\theta\cos\theta

$\theta$: Kofa $\text{rad}$

Yoom akka itti fayyadamtu

Gatii sine fi cosine kofa tokkoo fayyadamuun sine kofa dachaa shallaguuf fayyadami.

Of eeggadhu

Sin 2 theta jechuun 2 sin theta jechuu miti. Lama baay'isuu kan qabdu sine theta fi cosine theta lamaaniini.

Grade 10 Mathematics, fuula 187-237 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Pythagorean Trigonometric IdentityFoormulaa

\sin^2\theta + \cos^2\theta = 1

\sin^2\theta + \cos^2\theta = 1

$\theta$: Kofa $\text{rad}$

Yoom akka itti fayyadamtu

Gatii sine yookiin cosine beekame irraa isa kaan barbaaduuf walqixxoo kanatti fayyadami.

Of eeggadhu

Akka quadrant kofichi keessatti argamutti mallattoo poosatiivii yookiin negaatiivii sirrii ta'e murteessuu hin dagatin.

Grade 10 Mathematics, fuula 187-237 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Reference Angle Akka QuadrantGabatee wabiiQuadrant I · Quadrant II · Quadrant III · +1

Quadrant	Hanga Kofaa	Foomulaa Reference Angle
Quadrant I	$0^\circ < \theta < 90^\circ$	$\theta_R = \theta$
Quadrant II	$90^\circ < \theta < 180^\circ$	$\theta_R = 180^\circ - \theta$
Quadrant III	$180^\circ < \theta < 270^\circ$	$\theta_R = \theta - 180^\circ$
Quadrant IV	$270^\circ < \theta < 360^\circ$	$\theta_R = 360^\circ - \theta$

Yoom akka itti fayyadamtu

Kofa kamiyyuu kan iddoo dhaabbataa irra jiruuf kofa reference argachuuf gabatee kanatti fayyadami.

Of eeggadhu

Yeroo hunda x-axis sarara dalgaa (digirii 180 yookiin digirii 360) irraa hir'isi malee y-axis sarara gubbaa irraa miti.

Grade 10 Mathematics, fuula 187-237 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Seera sine, seera cosine fi foormulaa bal'inaaFoormulaa

a^{2} = b^{2} + c^{2} - 2bc\,\cos A

a^{2} = b^{2} + c^{2} - 2bc\,\cos A

$a,\,b,\,c$: dheerina cinaachawwan sadanii
$A,\,B,\,C$: rogawwan. Tokkoon tokkoon isaanii cinaacha qubee walfakkaataadhaan barreeffametti fuulleffata, kanaafuu A gara a fuulleffatee b fi c gidduu taa'a

Foormulaa	Yeroo itti fayyadamtu	Bakka itti hojjetu
$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$	Yeroo rogawwan lamaa fi cinaacha tokko beektu, yookaan cinaachawwan lamaa fi rog isaan keessaa tokkotti fuulleffate beektu.	Rog-sadee kamiyyuu keessatti.
$a^{2} = b^{2} + c^{2} - 2bc\,\cos A$	Yeroo cinaachawwan lamaa fi rog gidduu isaanii jiru beektuu fi cinaacha sadaffaa barbaaddu.	Rog-sadee kamiyyuu keessatti.
$\cos A = \dfrac{b^{2} + c^{2} - a^{2}}{2bc}$	Yeroo cinaachawwan sadanuu beektuu fi rog barbaaddu.	Rog-sadee kamiyyuu keessatti.
$\text{Area} = \tfrac{1}{2}\,ab\,\sin C$	Yeroo cinaachawwan lamaa fi rog gidduu isaanii jiru beektu. Rog-sadeen sunuma $\tfrac{1}{2}\,bc\,\sin A$ fi $\tfrac{1}{2}\,ac\,\sin B$ ni kenna.	Rog-sadee kamiyyuu keessatti.
$\sin A = \dfrac{\text{opposite}}{\text{hypotenuse}}$	Yeroo cinaacha rogichatti fuulleffatee fi hypotenuse beektu.	Rog-sadee rog sirrii qabu qofa keessatti.
$\cos A = \dfrac{\text{adjacent}}{\text{hypotenuse}}$	Yeroo cinaacha rogicha bira jiruu fi hypotenuse beektu.	Rog-sadee rog sirrii qabu qofa keessatti.
$\tan A = \dfrac{\text{opposite}}{\text{adjacent}}$	Yeroo cinaacha rogichatti fuulleffatee fi kan isa bira jiru beektu.	Rog-sadee rog sirrii qabu qofa keessatti.
$a^{2} + b^{2} = c^{2}$	Yeroo cinaachawwan lama beektuu fi sadaffaa barbaaddu. Asitti $c$ hypotenuse dha, kunis cinaacha rog sirriitti fuulleffatee dha.	Rog-sadee rog sirrii qabu qofa keessatti. Kunis seera cosine yeroo rogichi digrii 90 godhamu dha, sababii cosine isaa zeeroo ta'uuf.

Yoom akka itti fayyadamtu

Yeroo gaaffiin rog-sadee cinaachawwanii fi rogawwan walitti makuu fi rog-sadeen sun rog sirrii hin qabne itti fayyadami. Waan siif kenname gabatee waliin wal simsiisi: rogawwan lamaa fi cinaacha tokko seera sine barbaada; cinaachawwan lamaa fi rog gidduu isaanii jiru immoo seera cosine barbaada, walfakkaatuun sunis bal'ina kallattiidhaan siif kenna.

Of eeggadhu

Kiyyoowwan lama jiru, lamaanuu waa'ee rogni kam cinaachawwan kamiin akka deemuu ti. Seerri cosine baay'ata tokko qofa hir'isa: $2bc\,\cos A$ akka baay'ina tokkootti shallagii $b^{2} + c^{2}$ irraa hir'isi. Duraan $b^{2} + c^{2} - 2bc$ shallaganii ergasii $\cos A$ n baay'isuun yeroo hunda lakkoofsa adda ta'e kenna. Lammaffaa, rogichi kan cinaachawwan lamaan ati baay'iftu gidduu jiru ta'uu qaba. $\tfrac{1}{2}\,ab\,\sin C$ bal'ina kan ta'e sababii $C$ $a$ fi $b$ gidduu jiruuf qofa; $\cos A$ seera cosine keessaa immoo rog $b$ fi $c$ gidduu jiru dha. Qabxiin tokko dabalataa: yeroo seerri sine rog siif kennu, kaalkuleetarri deebii qara qofa deebisa, kanaafuu osoo hin fudhatin dura digrii 180 irraa rogicha hir'isuun kan argamu rog-sadee sanaaf ni ta'a moo hin ta'u mirkaneessi.

Gaaffii shaakalaa boqonnaa 4

Gaaffii filannoo 10 boqonnaa kana irratti barreeffaman, tokkoon tokkoon isaanii deebii fi sababa qabu. Deebii banuu dura yaali. Gaaffiin kun Ingiliffaa tiin.

Gaaffii 1

Convert 120° into radians.

A.2π/3
B.3π/2
C.π/3
D.4π/3

Deebii fi sababa agarsiisi

Deebii: A 2π/3

Sababa

Use the rule that π radian = 180°. So 120° = π × 120/180 = 2π/3.

Gaaffii 2

Convert 4π/3 radian into degrees.

A.180°
B.240°
C.270°
D.300°

Deebii fi sababa agarsiisi

Deebii: B 240°

Sababa

To change radians to degrees, multiply by 180/π. So 4π/3 × 180/π = 4 × 60 = 240°.

Gaaffii 3

What is the complementary angle of 57°?

A.33°
B.57°
C.123°
D.147°

Deebii fi sababa agarsiisi

Deebii: A 33°

Sababa

Two angles are complementary when they add up to 90°. So the complement of 57° is 90° - 57° = 33°.

Gaaffii 4

An angle x is in standard position. Sin x is positive and cos x is negative. In which quadrant is angle x?

A.First quadrant
B.Second quadrant
C.Third quadrant
D.Fourth quadrant

Deebii fi sababa agarsiisi

Deebii: B Second quadrant

Sababa

Sin x is positive in the first and second quadrants. Cos x is negative in the second and third quadrants. The only quadrant that fits both is the second quadrant.

Gaaffii 5

The point P(5, 12) is on the terminal side of angle θ in standard position. What is sin θ?

A.5/13
B.12/13
C.12/5
D.13/12

Deebii fi sababa agarsiisi

Deebii: B 12/13

Sababa

First find the distance r from the origin: r = √(5² + 12²) = √169 = 13. Then sin θ = y/r = 12/13.

Gaaffii 6

Find the reference angle θR for θ = 245°.

A.25°
B.55°
C.65°
D.115°

Deebii fi sababa agarsiisi

Deebii: C 65°

Sababa

245° is a third quadrant angle (between 180° and 270°). For a third quadrant angle, the reference angle is the angle minus 180°. So θR = 245° - 180° = 65°.

Gaaffii 7

Which of these angles is NOT co-terminal with 30°?

A.390°
B.-690°
C.750°
D.60°

Deebii fi sababa agarsiisi

Deebii: D 60°

Sababa

Co-terminal angles differ from 30° by a multiple of 360°. 390° = 30° + 360°. -690° = 30° - 720°. 750° = 30° + 720°. But 60° - 30° = 30°, which is not a multiple of 360°, so 60° is not co-terminal with 30°.

Gaaffii 8

A tall building in Addis Ababa is 40 m high. Abebe stands on top of it and sees a car on the level ground at the foot of the building. The angle of depression to the car is 60°. How far is the car from the base of the building?

A.40√3 m
B.40/√3 m
C.20√3 m
D.40 m

Deebii fi sababa agarsiisi

Deebii: B 40/√3 m

Sababa

The angle of depression equals the angle of elevation from the car, so the angle at the car is 60°. Using tan 60° = height/distance, we get distance = 40/tan 60° = 40/√3 m.

Gaaffii 9

Angle x is in the third quadrant and sin x = -3/5. What is cos x?

A.-4/5
B.4/5
C.-3/5
D.3/4

Deebii fi sababa agarsiisi

Deebii: A -4/5

Sababa

Use sin²x + cos²x = 1. So cos²x = 1 - (9/25) = 16/25, which gives cos x = ±4/5. In the third quadrant cos x is negative, so cos x = -4/5.

Gaaffii 10

The minute hand of a watch is 1.5 units long. How far does its tip move in 15 minutes? Use the rule l = rθ.

A.0.75π units
B.1.5π units
C.0.5π units
D.3π units

Deebii fi sababa agarsiisi

Deebii: A 0.75π units

Sababa

In 60 minutes the minute hand makes one full turn, so in 15 minutes it turns 15/60 = 1/4 of a turn. The angle is θ = (1/4) × 2π = π/2. Then l = r × θ = 1.5 × π/2 = 0.75π units.

Appiin boqonnaa kana irratti gaaffii 12 qaba; utuu hojjettuu ni sirreessa. Kan asii jiran 10 jalqabaa ti.