Su'aalo Tusaale ah oo Ka Hadlaya Saamiyada Trigonometric
Trigonometry waa laan xisaabeed oo ka hadlaysa xiriirka ka dhexeeya dhererka iyo xaglaha saddexagalka. Saamiyada Trigonometric waxay door muhiim ah ka ciyaaraan dhinacyo kala duwan sida injineernimada, fiisigiska, xiddigiska, iyo xitaa nolol maalmeedka. Maqaalkani wuxuu dib u eegi doonaa dhowr tusaale oo dhibaatooyin ah iyo xalalkooda si uu kaaga caawiyo inaad fahamto fikradaha aasaasiga ah ee saamiyada trigonometric.
Saamiyada Trigonometric ee Saddexagalada Midig
Aan ku bilowno fahamka saamiga aasaasiga ah ee saddexagalka midig. Saamiyadan waxaa loo yaqaan sine (sin), cosine (cos), iyo tangent (tan). Saddexagalka midig, waxaa jira saddex dhinac oo muhiim ah oo aan u baahanahay inaan aqoonsanno:
1. Dhinaca ka soo horjeeda: Dhinaca si toos ah uga soo horjeeda xagasha aan ka fiirsaneyno.
2. Dhinaca ku xiga: Dhinaca ku xiga xagasha aan ka fiirsanayno.
3. Hoos u dhac: Dhinaca ugu dheer ee saddexagalka midig ee ka soo horjeeda xagasha midig.
Su'aal Tusaale 1aad
Marka la eego saddexagal midig oo leh xagal θ, halkaas oo dhinaca ka soo horjeeda xagasha θ uu leeyahay dherer dhan 3 cutub, dhinaca ku xiga wuxuu leeyahay dherer dhan 4 cutub, halka hypotenuse-kuna uu leeyahay dherer dhan 5 cutub. Soo hel qiimaha sin, cos, iyo tan ee xagasha θ.
Xalka:
Si loo helo qiimaha sine, cosine, iyo tangent ee xagasha θ, waxaan isticmaalnaa qaacidooyinka saamiga trigonometric ee aasaasiga ah:
- Sine (dembiga) θ = Dhinaca hore / Hypotenuse
\[ \sin θ = \frac{3}{5} \]
– Cosine (cos) θ = Dhinac Dhinac / Hypotenuse
\[ \cos θ = \frac{4}{5} \]
– Tangent (tan) θ = Dhinaca Hore / Dhinaca
\[ \tan θ = \frac{3}{4} \]
Xiriirka Trigonometric ee Xaglaha Kale
Saamiyada Trigonometric waxaa sidoo kale lagu dabaqi karaa xaglo kale oo ku jira noocyo kala duwan oo saddexagal ah, oo ay ku jiraan saddexagal siman iyo saddexagal caadi ah. Waa muhiim in la fahmo qawaaniinta aasaasiga ah sida Xeerka Sine iyo Xeerka Cosine, kuwaas oo khuseeya saddexagal aan ahayn saddexagal sax ah.
Su'aal Tusaale 2aad
Saddex xagal ABC oo leh dhinacyo a = 7 cm, b = 24 cm, iyo xagal C = 90 °. Soo hel dhererka dhinaca c iyo qiimaha xaglaha A iyo B.
Xalka:
Maadaama xagasha C ay tahay xagal qumman, waxaan isticmaali karnaa Aragtida Pythagorean si aan u helno dhererka dhinaca c (hypotenuse):
\[ c = \sqrt{a^2 + b^2} \]
\[ c = \sqrt{7^2 + 24^2} \]
\[ c = \sqrt{49 + 576} \]
\[ c = \sqrt{625} \]
\[ c = 25 \, \qoraalka{cm} \]
Marka xigta, si loo go'aamiyo qiimaha xaglaha A iyo B, waxaan isticmaali karnaa saamiga trigonometric aasaasiga ah.
Xagasha A, waxaan u isticmaalnaa cosine:
\[ \cos A = \frac{\text{dhinaca ku xiga}}{\text{hypotenuse}} = \frac{24}{25} \]
\[ A = \cos^{-1} \left(\frac{24}{25}\right) \]
Xagasha B, waxaan isticmaalnaa sine-ka:
\[ \sin B = \frac{\text{obverse side}}{\text{hypotenuse}} = \frac{24}{25} \]
\[ B = \sin^{-1} \left(\frac{24}{25}\right) \]
Sababtoo ah saddexagal midig wadarta xaglaha aan ahayn xagasha saxda ah waa inay noqotaa 90°:
\[ A + B = 90° \]
\[ A = 90° – B \]
Xeerka Sine
Xeerka sine-ka waxaa lagu sheegay sidan soo socota:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
Su'aal Tusaale 3aad
Waxaa la siiyay saddexagal leh dhinac a = 8 cm, dhinac b = 15 cm, iyo xagal C = 60 °. Soo hel xagasha A iyo dhererka dhinaca c adoo isticmaalaya Xeerka Sines.
Xalka:
Marka hore, isticmaal Xeerka Sines si aad u hesho mid ka mid ah xaglaha kale (tusaale ahaan, xagasha A):
\[ \frac{a}{\sin A} = \frac{c}{\sin C} \]
\[ \frac{8}{\sin A} = \frac{c}{\sin 60°} \]
Waxaan u baahanahay inaan marka hore xisaabino qiimaha c. Maadaama xagasha C ay tahay 60°, waxaan isticmaali karnaa Xeerka Cosine:
\[ c^2 = a^2 + b^2 – 2ab \cos(C) \]
\[ c^2 = 8^2 + 15^2 – 2 \cdot 8 \cdot 15 \cdot \cos(60°) \]
\[ c^2 = 64 + 225 – 240 \cdot 0.5 \]
\[ c^2 = 289 – 120 \]
\[ c^2 = 169 \]
\[ c = 13 \, \qoraalka{cm} \]
Hadda, waxaan ka heli karnaa xagasha A:
\[ \frac{8}{\sin A} = \frac{13}{\sin 60°} \]
Tan iyo \(\sin 60° = \sqrt{3}/2\):
\[ \frac{8}{\sin A} = \frac{13}{\sqrt{3}/2} \]
\[ \frac{8}{\sin A} = \frac{26}{\sqrt{3}} \]
\[ 8 \cdot \sqrt{3} = 26 \sin A \]
\[ \sin A = \frac{8 \sqrt{3}}{26} \]
\[ \sin A = \frac{4 \sqrt{3}}{13} \]
Ugu dambeyntii, isticmaal habka loo yaqaan 'version sine':
\[ A = \sin^{-1}\left(\frac{4 \sqrt{3}}{13}\right) \]
Xeerka Cosine-ka
Xeerka cosine-ku wuxuu leeyahay:
\[ c^2 = a^2 + b^2 – 2ab \cos C \]
Su'aal Tusaale 4aad
Waxaa la siiyay saddexagal leh dhinac a = 9 cm, dhinac b = 12 cm, iyo dhinac c = 15 cm. Soo hel xagasha C.
Xalka:
Si aan u helno xagasha C, waxaan isticmaalnaa Xeerka Cosine:
\[ c^2 = a^2 + b^2 – 2ab \cos C \]
\[ 15^2 = 9^2 + 12^2 – 2 \cdot 9 \cdot 12 \cdot \cos C \]
\[ 225 = 81 + 144 - 216 \ cos C \]
\[ 225 = 225 – 216 \cos C \]
\[ 0 = -216 \cos C \]
\[ \cos C = 0 \]
Laga bilaabo halkan waxaan ognahay in xagasha C ay tahay 90° sababtoo ah cos 90° = 0.
Kuwani waa qaar ka mid ah dhibaatooyinka tusaale ahaan kaa caawinaya inaad fahamto saamiga trigonometric. Ku celcelinta joogtada ah ee dhibaatooyinka kala duwan waxay si weyn kaaga caawin doontaa inaad xoojiso fahamkaaga ku saabsan fikradahan. Waxaan rajeyneynaa in maqaalkani uu kaa caawiyay barashadaada!
