Answer :
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1° Without using a calculator:
a) For the question where cos x = 0.6, the range of values for cos x is between -1 and 1. Since 0.6 falls within this range, there exists an acute angle x where cos x = 0.6. Therefore, the answer is "yes."
b) When sin x = 0.9, the range for sin x is also between -1 and 1. Since 0.9 is within this range, there exists an acute angle x where sin x = 0.9. Hence, the answer is "yes."
c) For tan x = 0.78, tan x can take on any real value, so there exists an acute angle x where tan x = 0.78. Therefore, the answer is "yes."
2° If x exists:
a) To calculate x when cos x = 0.6, you can use the inverse cosine function. So, x = cos^(-1)(0.6) ≈ 53.13°.
b) To find x when sin x = 0.9, you can use the inverse sine function. Therefore, x = sin^(-1)(0.9) ≈ 64.35°.
c) When tan x = 0.78, you can calculate x using the inverse tangent function. Thus, x = tan^(-1)(0.78) ≈ 38.67°.
In the case where cos x = 1.5 or sin x = 1.4, these values are outside the range of possible values for cosine and sine functions, so there are no solutions for acute angles in these cases.
