Piv txwv cov lus nug thiab kev sib tham txog Trigonometry
Trigonometry yog ib ceg ntawm kev lej uas kawm txog kev sib raug zoo ntawm cov ces kaum thiab qhov ntev ntawm sab hauv cov duab peb ceg. Kev nkag siab txog cov ntsiab lus yooj yim ntawm trigonometry yog qhov tseem ceeb heev rau ntau yam kev siv, txij li physics thiab engineering mus rau astronomy thiab geography. Hauv tsab xov xwm no, peb yuav piav qhia ntau yam piv txwv teeb meem thiab cov lus piav qhia kom tiav los pab nkag siab.
Piv txwv lus nug 1: Xam cov sab ntawm lub duab peb sab siv txoj kev Sine
Lo lus nug:
Muab ib daim duab peb ceg ABC, nrog lub kaum sab A = 30°, lub kaum sab B = 45°, thiab sab b = 10 cm. Xam qhov ntev ntawm sab a.
Kev Sib Tham:
Siv txoj cai ntawm sines:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]
Sau cov nqi uas paub lawm:
\[ \frac{a}{\sin 30°} = \frac{10}{\sin 45°} \]
Peb paub tias:
\[ \sin 30° = \frac{1}{2} \]
\[ \sin 45° = \frac{2}{2} \]
Tam sim no, hloov cov nqi no rau hauv qhov sib npaug:
\[ \frac{a}{\frac{1}{2}} = \frac{10}{\frac{2}{2}} \]
Ua kom yooj yim dua qhov sib npaug:
\[ 2a = \frac{10 \cdot 2}{\sqrt{2}} \]
\[ 2a = \frac{20}{\sqrt{2}} \]
Ua kom tus lej sib npaug:
\[ 2a = \frac{20 \cdot \sqrt{2}}{2} \]
\[ 2a = 10\sqrt{2} \]
\[ ib = 5\sqrt{2} \]
Yog li, qhov ntev ntawm sab a yog 5\sqrt{2}\) cm.
Piv Txwv Lo Lus Nug 2: Xam Cov Ces kaum Siv Txoj Kev Cosine
Lo lus nug:
Ib daim duab peb ceg muaj sab a = 7 cm, b = 10 cm, thiab c = 5 cm. Txheeb xyuas qhov loj ntawm lub kaum sab C.
Kev Sib Tham:
Siv txoj cai ntawm cosines:
\[ c^2 = a^2 + b^2 – 2ab \cos C \]
Sau cov nqi uas paub lawm:
\[ 5^2 = 7^2 + 10^2 – 2 \cdot 7 \cdot 10 \cdot \cos C \]
Ua kom yooj yim dua qhov sib npaug:
\[ 25 = 49 + 100 – 140 \cos C \]
\[ 25 = 149 – 140 \cos C \]
Tsiv 149 mus rau sab laug:
\[ 25–149 = -140 \cos C \]
\[ -124 = -140 \cos C \]
\[ \cos C = \frac{124}{140} \]
\[ \cos C = \frac{62}{70} \]
\[ \cos C = \frac{31}{35} \]
Siv lub tshuab xam zauv los nrhiav \( \cos^{-1} \) (inverse cosine):
\[ C \approx \cos^{-1}\left(\frac{31}{35}\right) \]
\[ C \kwv yees li 25.84° \]
Yog li, qhov loj ntawm lub kaum sab xis C yog li ntawm 25.84 °.
Piv txwv lus nug 3: Xam qhov siab thiab cheeb tsam ntawm daim duab peb sab
Lo lus nug:
Ib daim duab peb ceg muaj ob sab ntev a = 6 cm thiab b = 8 cm nrog lub kaum sab xis ntawm lawv, \(\theta\) = 60°. Xam qhov siab thiab thaj tsam ntawm daim duab peb ceg.
Kev Sib Tham:
1. Xam Thaj Chaw ntawm Daim Duab Peb Sab:
Siv cov mis rau thaj tsam ntawm daim duab peb sab:
\[ \text{Cheeb Tsam} = \frac{1}{2} ab \sin \theta \]
Sau cov nqi uas paub lawm:
\[ \text{Cheeb Tsam} = \frac{1}{2} \cdot 6 \cdot 8 \cdot \sin 60° \]
Peb paub tias:
\[ \sin 60° = \frac{3}{2} \]
Yog li ntawd:
\[ \text{Cheeb Tsam} = \frac{1}{2} \cdot 6 \cdot 8 \cdot \frac{\sqrt{3}}{2} \]
\[ \text{Cheeb Tsam} = \frac{1}{2} \cdot 6 \cdot 8 \cdot \frac{\sqrt{3}}{2} \]
\[ \text{Cheeb Tsam} = 24 \cdot \frac{\sqrt{3}}{2} \]
\[ \text{Cheeb Tsam} = 12\sqrt{3} \]
Yog li, thaj tsam ntawm lub duab peb sab yog \( 12\sqrt{3} \) cm².
2. Xam Qhov Siab ntawm Ib Lub Duab Peb Sab los ntawm Lub Hauv Paus a:
Yuav xam qhov siab ntawm daim duab peb sab, qhia qhov siab ua h thiab siv cov mis cheeb tsam:
\[ \text{Cheeb Tsam} = \frac{1}{2} \times a \times h \]
\[ 12\sqrt{3} = \frac{1}{2} \times 6\times h \]
\[ 12\sqrt{3} = 3h \]
\[ h = \frac{12\sqrt{3}}{3} \]
\[ h = 4\sqrt{3} \]
Yog li, qhov siab ntawm daim duab peb sab yog \( 4\sqrt{3} \) cm.
Piv Txwv Lo Lus Nug 4: Txheeb Xyuas Cov Sab Ntawm Ib Lub Duab Peb Sab Xis
Lo lus nug:
Hauv ib daim duab peb sab xis, nrog lub kaum sab xis \(\theta\) = 30° thiab sab sib luag ntawm lub kaum sab xis \(\theta\) yog 5 cm, txiav txim siab qhov ntev ntawm lub hypotenuse.
Kev Sib Tham:
Siv cov piv txwv trigonometric rau lub kaum sab xis 30° hauv daim duab peb sab:
\[ \sin \theta = \frac{\text{front}}{\text{hypotenuse}} \]
\[ \sin 30° = \frac{obverse}{hypotenuse} = \frac{5}{\text{hypotenuse}} \]
Peb paub tias:
\[ \sin 30° = \frac{1}{2} \]
Yog li ntawd:
\[ \frac{1}{2} = \frac{5}{\text{hypotenuse}} \]
\[ \text{hypotenuse} = 10 \]
Yog li ntawd, qhov ntev ntawm lub hypotenuse yog 10 cm.
Piv txwv lus nug 5: Xam cov ces kaum nrog Trigonometric Functions
Lo lus nug:
Yog tias \( \tan \theta = \frac{3}{4} \), xam qhov loj ntawm lub kaum sab xis \(\theta\).
Kev Sib Tham:
Siv cov qauv inverse tangent:
\[ \theta = \tan^{-1} \left(\frac{3}{4}\right) \]
Nrog kev pab los ntawm lub tshuab xam zauv:
\[ \theta \approx 36.87° \]
Yog li, qhov loj ntawm lub kaum sab xis ∞(\theta\) yog li 36.87°.
Xaus
Trigonometry yog ib lub tswv yim lej dav dav uas siv tau rau ntau qhov chaw sib txawv. Kev nkag siab txog kev siv cov cai ntawm sines, cosines, thiab lwm yam kev ua haujlwm yooj yim ua rau peb daws tau ntau yam teeb meem uas cuam tshuam nrog cov duab peb ceg thiab cov ces kaum. Los ntawm cov piv txwv saum toj no, vam tias cov neeg nyeem yuav nkag siab tob dua thiab siv tau rau ntau qhov xwm txheej uas xav tau kev nkag siab txog trigonometry.
