Piv txwv ntawm Cov Lus Nug thiab Kev Sib Tham txog Cov Ces kaum Tshwj Xeeb hauv Trigonometric Ratios
Trigonometry yog ib ceg ntawm kev suav lej uas kawm txog kev sib raug zoo ntawm cov sab thiab cov ces kaum ntawm cov duab peb ceg. Ib lub tswv yim tseem ceeb hauv trigonometry yog kev siv cov ces kaum tshwj xeeb los nkag siab txog cov piv txwv trigonometric. Cov ces kaum tshwj xeeb uas siv ntau suav nrog 0°, 30°, 45°, 60°, thiab 90°. Tsab xov xwm no yuav piav qhia txog cov piv txwv thiab tham txog cov ces kaum tshwj xeeb hauv cov piv txwv trigonometric.
Kev Taw Qhia rau Cov Lub kaum sab xis tshwj xeeb
Cov ces kaum tshwj xeeb tau txais los ntawm kev tshuaj xyuas cov duab peb ceg tshwj xeeb, xws li isosceles thiab equilateral triangles. Nov yog cov nqi trigonometric yooj yim rau cov ces kaum tshwj xeeb kom nco ntsoov:
| Lub kaum sab xis (θ) | Sin(θ) | Cos(θ) | Tan(θ) |
|———–|——–|——–|——–|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | – |
Los ntawm kev paub cov nqi yooj yim no, peb tuaj yeem daws tau ntau yam teeb meem uas cuam tshuam txog trigonometric ratios ntawm cov ces kaum tshwj xeeb.
Cov Lus Nug Piv Txwv thiab Kev Sib Tham
Cia peb saib qee cov lus nug piv txwv thiab lawv cov kev sib tham:
Piv txwv lus nug 1
Lo lus nug:
Xam tus nqi ntawm \( \sin(30°) + \cos(60°) \).
Kev Sib Tham:
Peb siv cov nqi yooj yim ntawm lub kaum sab xis tshwj xeeb trigonometry.
\[
\sin(30°) = \frac{1}{2}
\]
\[
\cos(60°) = \frac{1}{2}
\]
Yog li ntawd,
\[
\sin(30°) + \cos(60°) = \frac{1}{2} + \frac{1}{2} = 1
\]
So, \( \sin(30°) + \cos(60°) = 1 \).
Piv txwv lus nug 2
Lo lus nug:
Txheeb xyuas tus nqi ntawm \( \tan(45°) \times \cos(45°) \).
Kev Sib Tham:
Peb siv cov nqi los ntawm lub rooj tshwj xeeb ntawm lub kaum sab xis.
\[
\tan(45°) = 1
\]
\[
\cos(45°) = \frac{\sqrt{2}}{2}
\]
Yog li ntawd,
\[
\tan(45°) \times \cos(45°) = 1 \times \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}
\]
Yog li, \( \tan(45°) \times \cos(45°) = \frac{\sqrt{2}}{2} \).
Piv txwv lus nug 3
Lo lus nug:
Yog tias \( \sin(θ) = \cos(θ) \), txiav txim siab tus nqi ntawm \( θ \) nyob rau hauv qhov ntau ntawm 0 ° txog 90 °.
Kev Sib Tham:
Los ntawm cov kev sib raug zoo ntawm trigonometry:
\[
\sin(θ) = \cos(θ)
\]
Qhov no txhais tau tias \( \tan(θ) = 1 \).
Tus nqi ntawm \( θ \) uas ua tiav qhov sib npaug \( \tan(θ) = 1 \) yog 45°.
Yog li, θ = 45°.
Piv txwv lus nug 4
Lo lus nug:
Xam tus nqi ntawm \( \frac{\sin(30°)}{\cos(60°)} \).
Kev Sib Tham:
Peb siv cov nqi los ntawm lub rooj tshwj xeeb ntawm lub kaum sab xis.
\[
\sin(30°) = \frac{1}{2}
\]
\[
\cos(60°) = \frac{1}{2}
\]
Yog li ntawd,
\[
\frac{\sin(30°)}{\cos(60°)} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1
\]
So, \( \frac{\sin(30°)}{\cos(60°)} = 1 \).
Piv txwv lus nug 5
Lo lus nug:
Txheeb xyuas tus nqi ntawm \( \cos(30°) \times \tan(60°) \).
Kev Sib Tham:
Peb siv cov nqi los ntawm lub rooj tshwj xeeb ntawm lub kaum sab xis.
\[
\cos(30°) = \frac{\sqrt{3}}{2}
\]
\[
\tan(60°) = \sqrt{3}
\]
Yog li ntawd,
\[
\cos(30°) \times \tan(60°) = \frac{\sqrt{3}}{2} \times \sqrt{3} = \frac{3}{2}
\]
Yog li, \( \cos(30°) \times \tan(60°) = \frac{3}{2} \).
Piv txwv lus nug 6
Lo lus nug:
Nrhiav tus nqi ntawm \(2 \sin(45°) \cos(45°) \).
Kev Sib Tham:
Peb siv cov nqi los ntawm lub rooj tshwj xeeb ntawm lub kaum sab xis.
\[
\sin(45°) = \frac{\sqrt{2}}{2}
\]
\[
\cos(45°) = \frac{\sqrt{2}}{2}
\]
Yog li ntawd,
\[
2 \sin(45°) \cos(45°) = 2 \times \frac{\sqrt{2}}{2} \times \frac{\sqrt{2}}{2} = 2 \times \frac{2}{4} = 1
\]
So, \(2 \sin(45°) \cos(45°) = 1 \).
Piv txwv lus nug 7
Lo lus nug:
Txheeb xyuas tus nqi ntawm \( \csc(30°) \).
Kev Sib Tham:
\( \csc(θ) \) is the inverse of \(\sin(θ) \).
\[
\sin(30°) = \frac{1}{2}
\]
Yog li ntawd,
\[
\csc(30°) = \frac{1}{\sin(30°)} = \frac{1}{\frac{1}{2}} = 2
\]
Yog li, \( \csc(30°) = 2 \).
Piv txwv lus nug 8
Lo lus nug:
Xam tus nqi ntawm \( \cot(60°) \).
Kev Sib Tham:
\( \cot(θ) \) is the inverse of \(\tan(θ) \).
\[
\tan(60°) = \sqrt{3}
\]
Yog li ntawd,
\[
\cot(60°) = \frac{1}{\tan(60°)} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}
\]
Yog li, \( \cot(60°) = \frac{3}{3} \).
Piv txwv lus nug 9
Lo lus nug:
Yog tias lub kaum sab xis uas tus nqi trigonometric yog sin(theta) = cos(45°)), nrhiav tus nqi ntawm 0° txog 90°.
Kev Sib Tham:
Los ntawm lub rooj tshwj xeeb lub kaum sab xis:
\[
\cos(45°) = \frac{\sqrt{2}}{2}
\]
Yog li ntawd,
\[
\sin(\theta) = \frac{2}{2}
\]
Nws yog paub,
\[
\sin(45°) = \frac{\sqrt{2}}{2}
\]
Yog li, \( \theta = 45° \).
Xaus
Kev paub txog cov ces kaum tshwj xeeb thiab cov nqi trigonometric yooj yim yog qhov tseem ceeb rau kev nkag siab txog cov ntsiab lus trigonometry thiab daws ntau yam teeb meem lej. Nrog kev xyaum kom zoo, kev nco qab lub rooj kaum tshwj xeeb yuav yooj yim dua, thiab kev daws cov teeb meem trigonometry yuav sai dua thiab ua haujlwm tau zoo dua.
Thaum kawg, tsab xov xwm no nthuav qhia qee qhov teeb meem piv txwv thiab kev sib tham txog cov ces kaum tshwj xeeb, pab koj nkag siab yuav ua li cas siv cov nqi trigonometric tshwj xeeb. Peb vam tias tsab xov xwm no tau pab tau koj txoj kev kawm!
