Example 1 :
Find the value of cos 105°.
Solution :
cos 105° = cos (60° + 45°)
cos (A + B) = cos A cos B - sin A sin B
cos (60 + 45) = cos 60° cos 45° - sin 60° sin 45° ----(1)
sin 45° = 1/√2 sin 60 = √3/2 |
cos 45° = 1/√2 cos 60° = 1/2 |
By applying the above values in the first equation, we get
cos (60 + 45) = (1/2) (1/√2) - (√3/2)(1/√2)
= (1/2√2) - (√3/2√2)
= (1 - √3)/2√2
So, the value of cos 105° is (1 - √3)/2√2.
Example 2 :
Find the value of sin 105°.
Solution :
sin 105° = sin (60 + 45)
sin (A + B) = sin A cos B - cos A sin B
sin (60 + 45) = sin 60° cos 45° + cos 60° sin 45° ----(1)
sin 60° = √3/2 sin 45° = 1/√2 |
cos 60° = 1/2 cos 45° = 1/√2 |
By applying the above values in the first equation, we get
sin (60 + 45) = (√3/2)(1/√2) + (1/√2)(1/2)
= (√3/2√2) + (1/2√2)
= (√3 + 1)/2√2
So, the value of sin 105° is (√3 + 1)/2√2.
Example 3 :
Find the value of tan 7π/12.
Solution :
tan 7π/12 = tan 105°
tan 105° = tan (60° + 45°)
tan (A + B) = (tan A + tan B) / (1 - tan A tan B)
= (tan 60 + tan 45)/(1 - tan 60 tan 45)
= (√3 + 1) / (1 - √3(1))
= (1 + √3) / (1 - √3)
Multiply the above fraction by its conjugate.
= [(1 + √3) / (1 - √3)] ⋅ [ (1 + √3) / (1 + √3)]
= (1 + √3)2 / (1 - √3) (1 + √3)
= (1 + 2√3 + 3) / (1 - 3)
= (4 + 2√3) / (-2)
= -2 (2 + √3)/ 2
= -(2 + √3)
So, the value of tan 7π/12 is -(2 + √3).
Kindly mail your feedback to v4formath@gmail.com
We always appreciate your feedback.
©All rights reserved. onlinemath4all.com
Apr 26, 24 01:51 AM
Apr 25, 24 08:40 PM
Apr 25, 24 08:13 PM