MATH 141 Lecture Notes - Lecture 13: Inverse Trigonometric Functions

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MATH 141 Lecture 13 SEPT 24
Inverse Trigonometric Functions
We want to restrict the domain of a trigonometric function so that it becomes one-one and then
we can define the inverse.
This is one-one, so it has inverse f’ = g in the Range (f) -> R
Since sin is continuous and sin 90 = +1 or -1, The Range (f) <= [-1,1]
Since function g has domain (g) = [-1,1], Range(g) = [-90,+90]
We set   
Properties of Inverse trigonometric functions:
       
  
  
  
   
Example: Simplify 
Solution:   
  
    
  
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Math 141 lecture 13 sept 24. We want to restrict the domain of a trigonometric function so that it becomes one-one and then we can define the inverse. This is one-one, so it has inverse f" = g in the range (f) -> r. Since sin is continuous and sin 90 = +1 or -1, the range (f) <= [-1,1] Since function g has domain (g) = [-1,1], range(g) = [-90,+90] We set n (cid:2869)(cid:1876)=(cid:1859)(cid:4666)(cid:1876)(cid:4667: (cid:1814) (cid:2778)(cid:2206)=(cid:2207) ,(cid:2188)(cid:2197)(cid:2200) (cid:2778)(cid:3409)(cid:2206)(cid:3409)(cid:2778, (cid:2206)=(cid:1814)(cid:2207) ,(cid:2188)(cid:2197)(cid:2200) (cid:2779)(cid:3409)(cid:2207)(cid:3409)(cid:2779, (cid:1814) (cid:2778)(cid:4666)(cid:1814)(cid:2206)(cid:4667)=(cid:2206) ,(cid:2188)(cid:2197)(cid:2200) (cid:2206)[ (cid:2779),(cid:2779), (cid:1814) (cid:4666)(cid:1814) (cid:2778)(cid:2206)(cid:4667)=(cid:2206) ,(cid:2188)(cid:2197)(cid:2200) (cid:2206) [ (cid:2778),(cid:2778)] Solution: (cid:1873)(cid:1868)(cid:1868)(cid:1867)(cid:1871)(cid:1857) =n (cid:2869) (cid:1876) ,(cid:1872) (cid:1857)(cid:1866) (cid:1872) (cid:1857) (cid:1870)(cid:1866)(cid:1859)(cid:1857) (cid:1867)(cid:1858) [ (cid:2870),(cid:2870)] (cid:1871)(cid:1866)(cid:1855)(cid:1857) (cid:1876)(cid:3410)(cid:882) (cid:1864)(cid:1871)(cid:1867) (cid:1866)(cid:1867)(cid:1872)(cid:1857) (cid:1872) (cid:1872)ec=(cid:883)\(cid:1855)(cid:1867)(cid:1871) (cid:1866)(cid:1856)n(cid:2870)+co(cid:2870)=(cid:883) (cid:1867)co= (cid:4666)(cid:883) n(cid:2870)(cid:4667)= (cid:883) ( (cid:1876))(cid:2870)= (cid:883) (cid:1876) (cid:2778) (cid:2778) (cid:2206)(cid:2779) =(cid:2778)\(cid:2185)(cid:2197)(cid:2201)= (cid:2778) (cid:2778) (cid:2206) (cid:1867)(cid:1875),co= (cid:4666)(cid:883) n(cid:2870)(cid:4667) (cid:1866)(cid:1855)(cid:1857) [ (cid:884),(cid:884)],co(cid:3410)(cid:882) (cid:1866)(cid:1856) (cid:1857)(cid:1866)(cid:1855)(cid:1857) (cid:1855)(cid:1867)(cid:1871)= (cid:4666)(cid:883) n(cid:2870)(cid:4667) Set (cid:1877)=n (cid:2869)(cid:1876) (cid:1857)(cid:1866)(cid:1855)(cid:1857)n(cid:1877)=(cid:1876) (cid:2186)(cid:2206)(cid:2186)(cid:2207)= (cid:2778)(cid:2186)(cid:2207)(cid:2186)(cid:2206)=(cid:2778)/(cid:4666)(cid:1815)(cid:2207)(cid:4667) ,(cid:2187)(cid:2196)(cid:2185)(cid:2187)(cid:2186)(cid:2186)(cid:2206)(cid:1814) (cid:2778)(cid:2206)= (cid:883) (cid:883) (cid:4666)(cid:1876)(cid:2873)(cid:4667)(cid:2870) . 5(cid:1876)(cid:2872)

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