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Given that 0 < x < 90° and sin(x+60°) = cos(2x), find the exact value of tan(x+20°)
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Your starting equation is
sin(x+60°) =cos(2x). (1)
Since cos(2x) = sin(90°-2x), you can re-write equation (2) in the next equivalent form
sin(x+60°) = sin(90°-2x). (2)
Since both angles, x+60° and 90°-2x are acute, equation (2) implies
x+60° = 90°-2x.
Solve it and find x
x + 2x = 90° - 60°,
3x = 30°,
x = 10°.
THEREFORE, tan(x+20°) = tan(10°+20°) = tan(30°) = .
ANSWER. If 0 < x < 90° and sin(x+60)° = cos(2x), then tan(x+20°) = .
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Found without using a calculator or tables, but exclusively
using the power of knowledge and intellect.
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But notice that, without using a calculator or tables, I found EXACT EXPRESSION
for tan(x+20°), ONLY - - - but not the value of tan(x+20°).
It is POSSIBLE to find exact expression without using calculator or tables,
but it is IMPOSSIBLE to find the VALUE without using a calculator or a table.
THEREFORE, the problem's formulation itself IS NOT CORRECT.
The correct formulation should ask to find EXACT EXPRESSION for tan(x+20°), but not the exact value of tan(x+20°).
Today, I saw SEVERAL similar posts from you for other expressions, also asking
to find exact value without a calculator or a table.
All these formulations are INCORRECT due to the same reason.
Correct formulations should ask to find exact expressions, but not about exact values.