Simplify: (help me please)?

sin(360-x) x tan(90-x)

1 Answer
Hammer
Jun 27, 2018

#=-cosx#

Explanation:

The sine function is periodic with period #360^@#, hence

#sin(360^@-x) =sin(-x)=-sinx#

About the tangent function; let #y=90^@ - x => x+y=90^@#.
#x# and #y# are complementary angles. We usually meet them in right triangles, where trigonometric functions are usually more familiar. As such, let #x# and #y# be the acute angles of a right triangle with sides #cc a, ccb, cc c#.

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#tanx="opposite"_x/"adjacent"_x=ccb/cca#

#cotx="adjacent"_x/"opposite"_x=cca/ccb#

#tany=tan(90^@-x)="opposite"_y/"adjacent"_y = cca/ccb=cotx#

#:. tan(90^@-x)=cotx#

Hence,

#sin(360^@-x)xxtan(90^@-x)=-sinx xx cotx=-sinx xx cosx/sinx=-cosx#

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