Problem 101 - Precalculus

Question

For the following exercises, state the reference angle for the given angle. $$ 135^{\circ} $$

Step-by-Step Solution

Answer

The reference angle for \(135^{\circ}\) is \(45^{\circ}\).

Step 1: Identify the Quadrant

The given angle is \(135^{\circ}\). To find the reference angle, first determine in which quadrant this angle lies. Angles from \(0^{\circ}\) to \(90^{\circ}\) lie in the first quadrant, from \(90^{\circ}\) to \(180^{\circ}\) in the second quadrant, from \(180^{\circ}\) to \(270^{\circ}\) in the third quadrant, and from \(270^{\circ}\) to \(360^{\circ}\) in the fourth quadrant. Since \(135^{\circ}\) is between \(90^{\circ}\) and \(180^{\circ}\), it lies in the second quadrant.

Step 2: Calculate Reference Angle

In the second quadrant, the reference angle is found by subtracting the given angle from \(180^{\circ}\). Thus, the reference angle for \(135^{\circ}\) is given by: \(180^{\circ} - 135^{\circ} = 45^{\circ}\).

Key Concepts

Quadrants in Trigonometry

In trigonometry, the Cartesian coordinate system is divided into four sections called quadrants. These quadrants are crucial for determining the sign and reference points for angles and trigonometric functions. Understanding the layout of these quadrants helps immensely when working with angles and trigonometric identities.

• First Quadrant: Angles range from \(0^{\circ}\) to \(90^{\circ}\). Here, all trigonometric functions are positive.
• Second Quadrant: Covers angles from \(90^{\circ}\) to \(180^{\circ}\). Only sine and cosecant functions are positive here.
• Third Quadrant: From \(180^{\circ}\) to \(270^{\circ}\). Tangent and cotangent functions are positive.
• Fourth Quadrant: Extends from \(270^{\circ}\) to \(360^{\circ}\). Cosine and secant are the positive functions in this quadrant.

The quadrants help determine the reference angle, which is the acute angle formed with the x-axis. Understanding in which quadrant an angle lies tells you not only about the sign of its trigonometric functions but also guides you to find its reference angle.

Angle Measurement

Angle measurement is fundamental in trigonometry, dealing with angles usually measured in degrees or radians. The standard way of measuring angles involves placing them within a circle, with the center point as the vertex of the angle.

• Degrees: This is the most common unit, with a complete circle measuring \(360^{\circ}\). Each quadrant of the circle accounts for \(90^{\circ}\).
• Radians: Often used in calculus and higher mathematics, a full circle is \(2\pi\) radians. This unit relates directly to the arc length of a circle.

Angles have both magnitude and direction; they are measured counter-clockwise from the positive x-axis. Recognizing these measurements helps identify which quadrant an angle falls into and thus their reference angle.

Trigonometric Identities

Trigonometric identities are mathematical equations that are true for all values of the variables within their domain. They play a crucial role in simplifying expressions and solving equations in trigonometry and related fields.

• Basic Identities: These include identities like \(\sin^2\theta + \cos^2\theta = 1\), which hold for any angle \(\theta\).
• Reciprocal Identities: These connect functions such as sine and cosecant: \(\csc\theta = \frac{1}{\sin\theta}\).
• Angle Sum and Difference: These involve formulas to find the sine, cosine, or tangent of two added or subtracted angles, e.g., \(\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b\).

By using these identities, one can often simplify complex trigonometric problems or verify the equivalence of different expressions. Understanding these identities allows for more profound insights into the behavior of trigonometric functions across different quadrants.