Problem 39 - Precalculus with Calculus Previews

Question

Find the given trigonometric function value. Do not use a calculator. $$ \sin 135^{\circ} $$

Step-by-Step Solution

Answer

\(\sin 135^{\circ} = \frac{\sqrt{2}}{2}\)

Step 1: Recognize the Reference Angle

To solve for \(\sin 135^{\circ}\), we start by finding its reference angle. Since 135° is in the second quadrant, we subtract it from 180°: 180° - 135° = 45°. So, the reference angle is 45°.

Step 2: Determine the Trigonometric Sign

In the second quadrant, the sine function is positive. Therefore, \(\sin 135^{\circ} = \sin 45^{\circ}\), and it remains positive.

Step 3: Use Known Angle Values

We know that \(\sin 45^{\circ} = \frac{\sqrt{2}}{2}\). Hence, by using the reference angle from Step 1, \(\sin 135^{\circ} = \frac{\sqrt{2}}{2}\).

Key Concepts

Reference Angle

The reference angle is a fundamental concept in trigonometry that helps simplify the calculation of trigonometric functions for angles greater than 90 degrees. The reference angle is the acute angle formed by the terminal side of the given angle and the x-axis. To find the reference angle for an angle in degrees:

• If the angle is in the second quadrant, subtract it from 180°.
• If the angle is in the third quadrant, subtract 180° from it.
• If the angle is in the fourth quadrant, subtract it from 360°.

In our example, the given angle is 135°. It's in the second quadrant. So, we subtract 135° from 180°, resulting in a reference angle of 45°. Understanding the reference angle is crucial because it helps us determine the exact value of trigonometric functions by relating them to the more familiar first quadrant.

Sine Function

The sine function, denoted as \( \sin \theta \), is one of the primary trigonometric functions and relates an angle in a right triangle to the ratio of the length of the opposite side over the length of the hypotenuse. For angles on the unit circle, the sine value is the y-coordinate of the point where the terminal side of the angle intersects the circle.Key properties of the sine function include:

• Range: The sine value is always between -1 and 1.
• Periodicity: The sine function repeats every 360° or \(2\pi\) radians.
• Symmetry: It is an odd function, meaning \(\sin(-\theta) = -\sin \theta\).

Knowing these properties helps us to predict the behavior of sine values over different angles and quadrants. Specifically, for \( \sin 45^{\circ} \), we already know \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \), which can be directly applied even in other quadrants by understanding the sine function's behavior there.

Angle Subtraction

Angle subtraction is a method used to determine a reference angle when dealing with angles outside the first quadrant. This process involves subtracting your angle from 90°, 180°, 270°, or 360° depending on the quadrant the angle lies in.For any angle \( \theta \) located in the second quadrant, like 135°, the subtraction process involves taking the difference between 180° and \( \theta \):\[\text{Reference Angle} = 180^{\circ} - \theta\]This subtraction helps to identify the equivalent acute angle, making it easier to find trigonometric values as these are often memorized in the first quadrant. It ensures that despite the angle's initial size or location, you can always revert to a simpler angle that shares the same trigonometric values, just adjusted for sign depending on the quadrant.

Second Quadrant

In the unit circle, the second quadrant is defined by angles ranging from 90° to 180°. This specific section of the circle impacts the sign of the trigonometric functions, specifically noting that:

• Sine values are positive.
• Cosine and tangent values are negative.

The significance of being in the second quadrant is that it affects the outcome of trigonometric evaluations. Even though the absolute value of \( \sin 135^{\circ} \) is \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \), the sign remains positive because the sine function is positive in the second quadrant. This knowledge is important when solving problems involving angles that are not in the first quadrant.

Unit Circle

The unit circle is a fundamental concept in trigonometry, consisting of a circle with a radius of one unit centered at the origin of the coordinate plane. It simplifies understanding of trigonometric functions since any angle's sine or cosine can be identified as the y or x coordinate, respectively, of the point where the terminal side of the angle intersects the circle.Key features include:

• The radius is always 1.
• Angles are typically measured from the positive x-axis.
• Sine corresponds to the y-coordinate, while cosine corresponds to the x-coordinate.

Using the unit circle, we find that for \( \sin 45^{\circ} \), the corresponding y-coordinate is \( \frac{\sqrt{2}}{2} \). Understanding the unit circle helps in visualizing how trigonometric functions behave over different angles, confirming predictions made by reference angles and quadrants.