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Example of finding the value of cosine double alfa

Question:

Given: tan a = 1/2, find the value of cos 2a.

Solution:

from definition: tan a = sin a/cos a
since tan a = 1/2 (given), so, sin a/cos a = 1/2
use the cross multiplication
so, cos a = 2 sin a
square both side of the equation
cos2a = 4 sin2a, name this as equation1
using the formula, sin2a + cos2a = 1, name this as equation2
substitute equation1 into equation2 to remove cos a
sin2a + 4 sin2a = 1
5 sin2a = 1
sin2a = 1/5
using the double angle formula,
cos2a = 1 - 2 sin2a
so, cos2a = 1 - 2 sin2a = 1 - 2 × 1/5 = 5/5 - 2/5 = 3/5

Therefore, cos2a = 3/5. Looking for more detail? Please watch the video.