Question

How do you find the exact value for `sin((3pi)/2)tan(pi/4) - cos((2pi)/3)`?

Answer 1

`-0.5`

Explanation:

A diagram Usually helps understanding

`sin(pi/4) = 1/sqrt(2)`

`cos(pi/4)= 1/sqrt(2)`

So`" "color(brown)( tan(pi/4)=sin(pi/4)/cos(pi/4) = sqrt(2)/(sqrt(2) )=1)`

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`color(brown)(sin(3/2 pi) = -sin(pi/2)=-1` ....Corrected

Given that the hypotenuse is of constant length:
As `theta` increases the hypotenuse gets steeper and steeper
dragging the opposite with it. In other words the opposite gets closer and closer to the point where you are measuring the angle.
This continues until you reach `pi/2` at which point the hypotenuse and the opposite coincide. Thus they are both of the same length. So `sin(pi/2)=1`

However, in the quadrant of `pi/2 " the " y"`value is negative

So `sin(3/2 pi)=-1`

`color(brown)(sin (3/2 pi)= -1)`

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`color(brown)(cos(2/3 pi) =-cos(1/3 pi) = -0.5`

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Given:`" "sin(3/2 pi)tan(pi/4)-cos(2/3 pi)`

`color(blue)((-1)(1) -(-0.5) = - 0.5)`