4.7

**Inverse Trigonometric Function** Sec 5π/3=2 à sec-1 2=? · To inverse a function, it must be one-to-one.The domain and range of the function will be the range and domain of itsinverse · y=sin-1 x *To make sin x one-to-one, we restrict the domain –π/2≤x≤π/2, and the range is [-1, 1] y=sin-1x D:[-1,1] R:[–π/2,π/2] Example: y=sin-1 (root2)/2  what is value of y Ordinarily, x can be π/4, 3π/4…But using inverse sin function, you must consider its range as well. Since the rrange is –π/2≤y≤π/2, the answer is π/4. ·  Cosine to be one-to-one, we restrict its domain [0,π] Y=Cos-1 x D: [-1, 1] R: [0,π] ·  Y= tan-1 x D: R R: ( –π/2, π/2) Ex: sin(-1) - ( root 3 /2)= -π/3   cos-1(root2)/2=-π/4 Cos-1 0=π/2 tan-1 -1= -π/4 Sin-1  <span style="font-size: 11pt; line-height: 115%; font-family: 'Calibri','sans-serif'; position: relative; top: 8.5pt; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA; mso-text-raise: -8.5pt;">  (-1/2)= <span style="font-size: 11pt; line-height: 115%; font-family: 'Calibri','sans-serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;">-π/6  tan-1- <span style="font-size: 11pt; line-height: 115%; font-family: 'Calibri','sans-serif'; position: relative; top: 8.5pt; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA; mso-text-raise: -8.5pt;"> (root3)= <span style="font-size: 11pt; line-height: 115%; font-family: 'Calibri','sans-serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;"> -π/3 <span style="font-family: Symbol; msofareastfontfamily: Symbol; mso-list: Ignore; msobidifontfamily: Symbol; msolist: Ignore;">· **More inverse:** Ex: csc-1   2/(root 3)= <span style="font-size: 11pt; line-height: 115%; font-family: 'Calibri','sans-serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;">π/3  csc 2/(root3)    =1.093 <span style="font-size: 11pt; line-height: 115%; font-family: 'Calibri','sans-serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;">+ cot-1-20=  tan-1 <span style="font-size: 11pt; line-height: 115%; font-family: 'Calibri','sans-serif'; position: relative; top: 8.5pt; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA; mso-text-raise: -8.5pt;"> <span style="msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast; mso-spacerun: yes; msospacerun: yes;">1/20= - 0.050 <span style="font-size: 11pt; line-height: 115%; font-family: 'Calibri','sans-serif'; position: relative; top: 8.5pt; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA; mso-text-raise: -8.5pt;">  R: (0,π) cot-1-20=-0.50+π=3.09 Find the exact value: cos-1 <span style="font-size: 11pt; line-height: 115%; font-family: 'Calibri','sans-serif'; position: relative; top: 8.5pt; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA; mso-text-raise: -8.5pt;"> is an angle that has cosine ¾. So the opposite side is <span style="font-size: 11pt; line-height: 115%; font-family: 'Calibri','sans-serif'; position: relative; top: 4.5pt; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA; mso-text-raise: -4.5pt;"> (root7). So the sin of that angle is (root7)/4. Homework: page 401 #2-26 multiple of 3 + Page 411 #3-33multiplt of 3 page 421 #1-20 all Elternative way: [|thi nguyen.docx]
 * y=csc-1 x D:(-∞,-1] U[1,∞) R: [–π/2, π/2],y≠0
 * y=sec-1 x D:(-∞,-1] U[1,∞) R:[0,π], y≠π/2
 * y=cot-1 x D: R R: (0,π)
 * Cos-1(cos -1.1)=1.1
 * Sin (cos-1 <span style="font-size: 11pt; line-height: 115%; font-family: 'Calibri','sans-serif'; position: relative; top: 8.5pt; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA; mso-text-raise: -8.5pt;"> <span style="msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast; mso-spacerun: yes; msospacerun: yes;">(3/4)=?