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Chapter 4: Problem 74

In Exercises \(61-86,\) use reference angles to find the exact value of each expression. Do not use a calculator. $$\sin \left(-225^{\circ}\right)$$

Short Answer

Expert verified
The exact value of \(\sin \left(-225^{\circ}\right)\) is \(\frac{\sqrt{2}}{2}\)

Step by step solution

01

Determine the position of the angle

The given angle is \(-225^{\circ}\). Trigonometric functions have a period of \(360^{\circ}\) or \(2\pi \) radian, so they repeat the same set of values for every full rotation. For an angle greater than \(180^{\circ}\), you can subtract \(360^{\circ}\) until you get a negative acute angle in the fourth quadrant. So, \(-225^{\circ} + 360^{\circ} = 135^{\circ}\), therefore \(-225^{\circ} \) and \(135^{\circ}\) are coterminal, meaning that they are in the same place on the unit circle.
02

Determine the reference angle

A reference angle for any angle is the acute angle to the x-axis. Even though \(-225^{\circ} \) and \(135^{\circ}\) both land on the same point in the unit circle, their reference angles are still \(45^{\circ}\) in either case. Note that for any angle, the reference angle is always positive.
03

Determine the sign of the trigonometric function

The reference angle sits in the second quadrant. As you may recall, sine is positive in the second quadrant.
04

Find the exact value using the reference angle

The reference angle already places the angle in the context of a right triangle, and now you simply find the sine of \(45^{\circ}\). That's an angle we know the sine of: \(\sin(45^{\circ}) = \frac{1}{\sqrt{2}}\) or \(\sin(45^{\circ}) = \frac{\sqrt{2}}{2}\), if you rationalize the denominator.

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