Problem 1 - Algebra and Trigonometry with Analytic Geometry

Question

Find the reference angle \(\theta_{\mathrm{k}}\) if \(\theta\) has the given measure. (a) \(240^{\circ}\) (b) \(340^{\circ}\) (c) \(-202^{\circ}\) (d) \(-660^{\circ}\)

Step-by-Step Solution

Answer

(a) \(60^{\circ}\), (b) \(20^{\circ}\), (c) \(22^{\circ}\), (d) \(60^{\circ}\).

Step 1: Understanding Reference Angles

The reference angle of any given angle \(\theta\) is the acute angle \(\theta_k\) formed by the terminal side of \(\theta\) and the horizontal axis. It is always measured positively as an angle less than \(90^\circ\).

Step 1: Finding Reference Angle for (a) \(240^{\circ}\)

Since \(240^{\circ}\) is in the third quadrant, subtract \(180^{\circ}\) to find the reference angle. \[ \theta_k = 240^{\circ} - 180^{\circ} = 60^{\circ} \]

Step 2: Finding Reference Angle for (b) \(340^{\circ}\)

Since \(340^{\circ}\) is in the fourth quadrant, subtract it from \(360^{\circ}\). \[ \theta_k = 360^{\circ} - 340^{\circ} = 20^{\circ} \]

Step 3: Finding Reference Angle for (c) \(-202^{\circ}\)

Add \(360^{\circ}\) to \(-202^{\circ}\) to find its positive coterminal angle. \(-202^{\circ} + 360^{\circ} = 158^{\circ}\). Since this is in the second quadrant, subtract \(158^{\circ}\) from \(180^{\circ}\). \[ \theta_k = 180^{\circ} - 158^{\circ} = 22^{\circ} \]

Step 4: Finding Reference Angle for (d) \(-660^{\circ}\)

Add \(360^{\circ}\) twice to \(-660^{\circ}\) to find a positive coterminal angle. \(-660^{\circ} + 720^{\circ} = 60^{\circ}\). Since \(60^{\circ}\) is already an acute angle, it is the reference angle. \[ \theta_k = 60^{\circ} \]

Key Concepts

Coterminal Angles

Coterminal angles are angles that share the same terminal side and differ from each other by full rotations, which are multiples of 360 degrees. Understanding coterminal angles is vital when dealing with angles outside the standard ( angle measurement range, due to their ability to bring angles back into a manageable scope.

• To find a coterminal angle, you can either add or subtract 360 degrees repeatedly.
• It helps in determining positive angles from negative ones, by allowing normalization within a 0-360 degree range.

If you start with an angle like -202 degrees, you can add 360 degrees to bring it to a positive angle, yielding 158 degrees in this case. This process helps when calculating the reference angle or understanding the position of an angle on a coordinate plane.

Quadrants in Trigonometry

In trigonometry, the coordinate plane is divided into four sections known as quadrants. These quadrants help in determining the sign of trigonometric functions based on the angle's position.

• The first quadrant ranges from 0 to 90 degrees and hosts positive trigonometric functions.
• The second quadrant spans from 90 to 180 degrees, with sine being positive.
• The third quadrant runs from 180 to 270 degrees, where tangent is positive.
• The fourth quadrant extends from 270 to 360 degrees, with cosine being positive.

Knowing which quadrant an angle lies in can help determine its reference angle, as each quadrant has a specific formula for calculating it. For instance, 240 degrees lie in the third quadrant, where you find the reference angle by subtracting 180 degrees. This understanding is crucial for solving trigonometric problems effectively.

Acute Angles

An acute angle is any angle less than 90 degrees, and it plays an important role in trigonometry. All reference angles are acute by definition, as they provide a way to simplify trigonometric calculations.
When determining the reference angle, ensure that the calculated angle does not exceed 90 degrees. If an angle like 60 degrees or 22 degrees appears, it's directly the reference angle since both of these values are less than 90 degrees already.
Acute angles offer simplified calculations because their trigonometric values are often intuitive or easier to determine, allowing for faster computations in trigonometric equations.

Angle Measurement Conversion

Angle measurement can be expressed in either degrees or radians, and knowing how to convert between the two is essential. Understanding how to shift an angle in degrees to radians and back helps in different mathematical contexts.

• The formula to convert degrees to radians is: \( ext{Radians} = ext{Degrees} \times \frac{\pi}{180} \).
• Conversely, to convert radians to degrees: \( ext{Degrees} = ext{Radians} \times \frac{180}{\pi} \).

While working through trigonometric problems, sometimes angles need to be expressed in radians, especially in calculus or advanced contexts. Practicing these conversions reinforces understanding and application in various mathematical fields, ensuring you can fluently navigate between angle measures.