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Values Involving Inverse Trigonometric Functions II

Values Involving Inverse Trigonometric Functions II. Finding arctrg(trgθ). Examples I. Arcsin. arcsin[sin( - π/ 4 )] = - π/ 4 Notice that: - π/ 4 belongs to the range of the function y = arcsinx, which is [ -π/2 , π/2 ]. Arctan. arctan[tan( - π/ 4 )] = - π/ 4

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Values Involving Inverse Trigonometric Functions II

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  1. Values Involving Inverse Trigonometric Functions II Finding arctrg(trgθ)

  2. Examples I

  3. Arcsin arcsin[sin( - π/ 4 )] = - π/ 4 Notice that: - π/ 4 belongs to the range of the function y = arcsinx, which is [ -π/2 , π/2 ]

  4. Arctan arctan[tan( - π/ 4 )] = - π/ 4 Notice that: - π/ 4 belongs to the range of the function y = arctanx, which is ( -π/2 , π/2 )

  5. Arccos arccos[cos( 3π/ 4 )] = 3π/ 4 Notice that: 3π/ 4 belongs to the range of the function y = arccosx, which is [ 0 , π ]

  6. Arccot arccot[cot( 3π/ 4 )] = 3π/ 4 Notice that: 3π/ 4 belongs to the range of the function y = arccotx, which is ( 0 , π )

  7. Arcsec arcsec[sec( 5π/ 4 )] = 5π/ 4 Notice that: 5π/ 4 belongs to the range of the function y = arcsecx, which is [ 0 , π/2 ) U [π , 3π/2 )

  8. Examples II

  9. Arcsin arcsin[sin( 3π/ 4 )] Notice that: 3π/ 4 does not belong to the range of the function y = arcsinx, which is [ -π/2 , π/2 ] sin( 3π/ 4 ) = 1/√2 ( Why? )Thus,arcsin[sin( 3π/ 4 )] = arcsin[1/√2] = π/ 4

  10. Arctan arctan[tan( 7π/ 4 )] Notice that: 7π/ 4 does not belong to the range of the function y = arctanx, which is ( -π/2 , π/2 ) tan( 7π/ 4 ) = -1 ( Why? )Thus,arctan[tan( 7π/ 4 )] = arcsin[-1] = -π/ 4

  11. Arccos arccos[cos( - π/ 4 )] Notice that: -π/ 4 does not belong to the range of the function y = arccosx, which is [ 0 , π ] cos( - π/ 4 ) = 1/√2 ( Why? )Thus,arccos[cos( -π/ 4 )] = arccos[1/√2] = π/ 4

  12. Arccot arccot[cot( 7π/ 4 )] Notice that: 7π/ 4 does not belong to the range of the function y = arccotx, which is ( 0 , π ) cot( 7π/ 4 ) = -1 ( Why? )Thus,arccot[cot( 7π/ 4 )] = arccot[-1] = 3π/ 4

  13. Arcsec arcsec[sec( 3π/ 4 )] Notice that:3π/ 4 does not belong to the range of the function y = arcsecx, which is [ 0 , π/2 )U[π,3π/2) sec( 3π/ 4 ) = - √2 ( Why? )Thus,arcsec[sec( 3π/ 4 )] = arcsec[ - √2] = 5π/ 4

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