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-135° is in quadrant 3 (going clockwise rather than counter-clockwise as we usually do). The reference angle is 45° degrees (the terminal arm is 45° from the x-axis).
To find the exact value, use the special triangles. sin(45°)=√2÷2 (rationalized form of 1÷√2, which you can get from the triangle: it has two sides of value 1, and by the Pythagoras theorem we get √2 as the hypotenuse; SOHCAHTOA gives us 1÷√2 for the sine value).The exact value of sin(-135°) would be -(√2÷2). (-135° is equivalent to 225°. sin(225°) does give the same answer as the negative of sin(45°)).
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We will see that the exact value is √2/2 = 0.707, and the reference angle is 45°.

How to determine the reference angle?

We know that for any angle θ we have that:

sin(θ) = sin(θ + n*360°)

Where n is an integer value.

To find the reference angle of a given angle θ, we just need to find a value of n such that:

0 ≤  θ + n*360° ≤ 360°

So we can write our angle in the normal range.

in this case, our angle is -135°

Then we can write:

0 ≤  -135° + n*360° ≤ 360°

Is immediate to notice that we must take n = 1

0 ≤  -135° + 1*360° ≤ 360°

0 ≤  -135° + 1*360° ≤ 360°

0 ≤  225° ≤ 360°

Now we know that θ = 225°.

Ok, this lies on the third quadrant, when an angle lies on the third quadrant, to get the reference angle you just need to subtract 180°.

We will get:

225° - 180° = 45°

So the reference angle is 45°.

Finally, to get the exact value we use:

sin (-135°) = sin(225°) = cos(45°) = √2/2 = 0.707

If you want to learn more about trigonometry, you can read:

https://brainly.com/question/8120556

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