Problem 10 - Algebra and Trigonometry

Question

9–32 Find the exact value of the trigonometric function. $$\sin 225^{\circ}$$

Step-by-Step Solution

Answer

\(-\frac{\sqrt{2}}{2}\)

Step 1: Understand the angle in standard position

The angle given is \(225^{\circ}\), which is in standard position. We need to determine which quadrant it lies in. \(225^{\circ}\) lies in the third quadrant since it is greater than \(180^{\circ}\) and less than \(270^{\circ}\).

Step 2: Reference angle calculation

The reference angle is the angle the given angle makes with the x-axis. For an angle in the third quadrant, the reference angle \( \theta' \) is calculated as \( \theta - 180^{\circ} \). Therefore, \( \theta' = 225^{\circ} - 180^{\circ} = 45^{\circ} \).

Step 3: Determine the function value

Since \(225^{\circ}\) is in the third quadrant, and we know the sine function is negative in this quadrant, we use the reference angle and change the sign accordingly. Given \( \sin(45^{\circ}) = \frac{\sqrt{2}}{2} \), it follows that \( \sin(225^{\circ}) = -\frac{\sqrt{2}}{2} \).

Key Concepts

Understanding the Sine Function

The sine function is one of the basic trigonometric functions. It is used to relate the angle in a right triangle to the ratio of the opposite side over the hypotenuse. Simply put:

• If you consider a right-angled triangle, the sine of an angle is the length of the side opposite the angle divided by the length of the hypotenuse.
• The sine function is periodic, meaning it repeats its values in regular intervals. Its period is every 360 degrees or \(2\pi\) radians.
• The sine function ranges from -1 to 1, going through all the values between negative and positive peaks in its full cycle.

When dealing with angles beyond 0 and 90 degrees, sine values can become negative, depending on the angle's location within the coordinate plane. This characteristic arises from how the sine of the angle is defined with respect to the unit circle.
Sine has unique properties in each quadrant, making it crucial to understand where the angle falls when determining its sine value.

Understanding the Reference Angle

A reference angle is the smallest angle formed between the terminal side of the original angle and the x-axis. It always helps simplify complex angle calculations by relating them to one of the basic trigonometric angles within the first quadrant.In calculations:

• Reference angles are always between 0 and 90 degrees, even if the original angle itself is not.
• When dealing with angles in the third quadrant, like \(225^\circ\), the reference angle is found by subtracting \(180^\circ\) from the original angle. For example, \(225^\circ - 180^\circ = 45^\circ\).

The reference angle lets us make use of known values of trigonometric functions for angles within the first quadrant. It acts as a shortcut, especially since sine, cosine, and tangent ratios of reference angles are the same as those in the first quadrant but with proper signs reflecting the angle's actual quadrant positioning.

Exploring the Third Quadrant

The third quadrant of the unit circle represents angles that are greater than \(180^\circ\) and up to \(270^\circ\). It's crucial to understand the properties of this quadrant to solve trigonometric problems like finding \(\sin 225^\circ\).In the third quadrant:

• Both sine and cosine functions are negative. This is due to both the x-coordinate (cosine) and the y-coordinate (sine) being negative in that region of the unit circle.
• The tangent function, however, is positive, since it is the ratio of sine to cosine, and dividing two negative numbers results in a positive number.

When evaluating the sine of an angle residing in the third quadrant using its reference angle, remember to change the sign to negative. For instance, the sine of \(225^\circ\) uses the reference angle \(45^\circ\), which typically gives \(\frac{\sqrt{2}}{2}\). But because \(225^\circ\) is in the third quadrant, we make it \(-\frac{\sqrt{2}}{2}\). This reflects how the sine function's behavior changes between different quadrants.