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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