How do you use double angle formulas to calculate cos 2x and sin 2x without finding x if #cos x = 3/5# and x is in the first quadrant?
2 Answers
Explanation:
The double-angle formulas state that:
Now, we are given that
Thus, using the identities mentioned above,
# cos2x = -7/25 #
# sin2x = 24/25 #
Explanation:
We have
Using
#sin^2x + (3/5)^2 = 1 #
# => sin^2x = 1-9/25 #
# :. \ sin^2x = 16/25 #
# :. \ \ sinx = +-sqrt(16/25) #
And knowing that
# => sinx = 4/5 #
Next we use the sine and cosine double angle formula, so that:
# cos2x -= cos^2x - sin^2 x #
# \ \ \ \ \ \ \ \ \ = (3/5)^2 - (4/5)^2 #
# \ \ \ \ \ \ \ \ \ = 9/25 - 16/25 #
# \ \ \ \ \ \ \ \ \ = -7/25 #
And:
# sin2x -= 2sinxcosx #
# \ \ \ \ \ \ \ \ \ = 2(3/5)(4/5) #
# \ \ \ \ \ \ \ \ \ = 24/25 #