In the previous section, we saw the fourth and fifth properties of inverse trigonometric functions. In this section, we will see the sixth property.
Property VI
• This has 3 parts:
(i)2tan−1x=sin−12x1+x2,|x|≤1(ii)2tan−1x=cos−11−x21+x2,x≥0(iii)2tan−1x=tan−12x1−x2,−1<x<1
• Let us prove part (i):
◼ Remarks:
• In line A, we assume that tan−1(x) = y
• In line B, we use the identity 15.
♦ List of identities can be seen here.
• In line C, we substitute for y from A.
• Let us prove part (ii):
◼ Remarks:
• In line A, we assume that tan−1(x) = y
• In line B, we use the identity 14.
♦ List of identities can be seen here.
• In line C, we substitute for y from A.
• Let us prove part (iii). We already saw the proof in property V. Here we will see an alternate method:
◼ Remarks:
• In line A, we assume that tan−1(x) = y
• In line B, we use the identity 16.
♦ List of identities can be seen here.
• In line C, we substitute for y from A.
Now we will see some solved examples.
Solved example 18.3
Show that
(i)sin−1(2x√1−x2)=2sin−1x,−1√2≤x≤1√2(ii)sin−1(2x√1−x2)=2cos−1x,−1√2≤x≤1
Solution:
Part (i):
◼ Remarks:
• Line 1: We assume that sin−1(x) = y
• Line 3: We use the identity sin2𝜃 + cos2𝜃 = 1.
• Line 6: We use the identity 15.
♦ List of identities can be seen here.
• Line 8: We take the inverse.
• Line 9: We substitute for y based on the assumption in (1).
Part (ii):
◼ Remarks:
• Line 1: We assume that cos−1(x) = y
• Line 3: We use the identity sin2𝜃 + cos2𝜃 = 1.
• Line 6: We use the identity 15.
♦ List of identities can be seen here.
• Line 8: We take the inverse.
• Line 9: We substitute for y based on the assumption in (1).
Solved example 18.4
Show that tan−112+tan−1211=tan−134
Solution:
◼ Remarks:
• Line 1: We use part 2 of property 5.
• Line 2: We substitute the known values.
In the next section, we
will see a few more solved examples.
Previous
Contents
Copyright©2024 Higher secondary mathematics.blogspot.com
No comments:
Post a Comment