Problem 86 - A Graphical Approach to Precalculus with Limits

Question

Use a reference angle to find \(\sin \theta\) and \(\cos \theta\) for the given \(\theta\). $$\theta=135^{\circ}$$

Step-by-Step Solution

Answer

\(\sin 135^{\circ} = \frac{\sqrt{2}}{2}\) and \(\cos 135^{\circ} = -\frac{\sqrt{2}}{2}\).

Step 1: Determine Reference Angle

Find the reference angle for \( \theta = 135^{\circ} \). The reference angle \( \theta' \) is the acute angle that \( \theta \) makes with the horizontal axis. Since \( 135^{\circ} \) is in the second quadrant, \( \theta' = 180^{\circ} - 135^{\circ} = 45^{\circ} \).

Step 2: Calculate Sine Using Reference Angle

Use the reference angle to find \( \sin \theta \) for \( \theta = 135^{\circ} \). Since \( 135^{\circ} \) is in the second quadrant, where sine is positive, the sine value is \( \sin 135^{\circ} = \sin 45^{\circ} = \frac{\sqrt{2}}{2} \).

Step 3: Calculate Cosine Using Reference Angle

Use the reference angle to find \( \cos \theta \) for \( \theta = 135^{\circ} \). In the second quadrant, cosine is negative, thus \( \cos 135^{\circ} = -\cos 45^{\circ} = -\frac{\sqrt{2}}{2} \).

Key Concepts

Reference Angle

A reference angle is a fundamental concept in trigonometry often used to simplify the calculation of trigonometric functions. It's defined as the acute angle formed between the terminal side of the given angle and the horizontal axis. Understanding the reference angle makes it easier to calculate trigonometric values because angles in different quadrants can be reduced to those in the first quadrant, where they hold the same trigonometric identities, albeit with different signs. To find a reference angle:

• If the angle is in the first quadrant, the reference angle is the angle itself.
• If the angle is in the second quadrant, subtract the angle from \(180^{\circ}\).
• For the third quadrant, subtract \(180^{\circ}\) from the angle.
• In the fourth quadrant, subtract the angle from \(360^{\circ}\).

For \(\theta = 135^{\circ}\), which lies in the second quadrant, the reference angle is calculated as follows: \(\theta' = 180^{\circ} - 135^{\circ} = 45^{\circ}\). This indicates that \(135^{\circ}\) shares the sine and cosine values of \(45^{\circ}\), but with signs appropriate for their quadrant.

Sine Function

The sine function is one of the three primary functions in trigonometry, usually defined in terms of a right triangle. The sine of an angle in a triangle is the length of the opposite side divided by the hypotenuse. In the context of reference angles and circular functions, the sine function provides the y-coordinate of a point on the unit circle. For any angle located in different quadrants:

• In the first quadrant, sine values are positive.
• In the second quadrant, sine remains positive.
• In the third and fourth quadrants, sine values are negative.

For \(\theta = 135^{\circ}\): Since it falls in the second quadrant, where the sine function is positive, the sine of \(135^{\circ}\) is equal to the sine of its reference angle, \(45^{\circ}\). Therefore, \(\sin 135^{\circ} = \sin 45^{\circ} = \frac{\sqrt{2}}{2}\). This maintains the value obtained from the unit circle, but the sign is determined by the quadrant.

Cosine Function

The cosine function is another trigonometric function, which is essential for understanding relationships in triangles. Cosine is defined as the ratio of the adjacent side of the angle to the hypotenuse in a right triangle.On the unit circle, cosine corresponds to the x-coordinate of a point. For angles beyond \(90^{\circ}\), reference angles help determine the cosine by understanding their location around the circle.Each quadrant changes the sign of cosine values:

• First quadrant: Cosine is positive.
• Second quadrant: Cosine is negative.
• Third quadrant: Cosine is negative.
• Fourth quadrant: Cosine is positive.

For \(\theta = 135^{\circ}\), positioned in the second quadrant where cosine is negative, it shares the magnitude with its reference angle. Thus, \(\cos 135^{\circ} = -\cos 45^{\circ} = -\frac{\sqrt{2}}{2}\). The reference angle gives us the cosine value, and the sign adjustment is a quick way to account for the angle's placement on the circle.