Problem 24 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Describe and correct the error in finding a reference angle \(\theta^{\prime}\) for \(\theta=650^{\circ}\).

Step-by-Step Solution

Answer

The reference angle \(\theta^{\prime}\) for \(\theta=650^\circ\) is 70 degrees.

Step 1: Reduce the Angle

First, let's reduce the given angle \(650^{\circ}\) to an equivalent angle that lies between 0 and 360 degrees. We can do that by subtracting 360 from 650 until we get a value less than 360. This is because a full circle is \(360^{\circ}\) and any angle with a measure greater than 360 degrees wraps around the circle more than once. So the equivalent angle to \(650^{\circ}\) is \(650-360=290^{\circ}\).

Step 2: Find the Reference Angle

Now once we have an angle that is less than \(360^{\circ}\), the next step is to find the reference of the angle \(290^{\circ}\). The reference angle (\(\theta^{\prime}\)) for an angle \(\theta\) between \(180^{\circ}\) and \(360^{\circ}\) is calculated as \(\theta^{\prime} = 360 - \theta\). So for \(\theta = 290^{\circ}\), the reference angle \(\theta^{\prime}\) would be \(360 - 290 = 70^{\circ}\).

Key Concepts

Angle Reduction

Angle reduction is a key technique in trigonometry that simplifies complex angles into more manageable ones. When given an angle, like 650 degrees, you might notice it's much larger than 360 degrees, the total degrees in a full circle. Angles greater than 360 degrees wrap around the circle multiple times, repeating themselves every 360 degrees. This is similar to how the hands of a clock repeat every 12 hours.

To reduce the angle, you subtract 360 degrees until the angle falls within the 0 to 360-degree range. For 650 degrees:

• Subtract 360 once: 650 - 360 = 290 degrees.

This reduced angle, 290 degrees, is equivalent to 650 degrees within a basic circle, meaning it points to the same position on the circle's circumference. Angle reduction helps in finding reference angles and evaluating trigonometric functions efficiently.

Degrees and Radians

Degrees and radians are two units of measurement for angles. While degrees are often more familiar, especially in daily contexts, radians are the standard in many mathematical computations, notably in calculus and advanced trigonometry.

Here’s a quick comparison:

• Degrees: A circle is 360 degrees. This unit is intuitive and used in many practical situations, such as navigation and surveying.
• Radians: A circle is 2π radians. This unit is derived from the radius of the circle and is preferred in academic and scientific computations because it simplifies many equations and formulas.

To convert from degrees to radians, use the conversion formula:\[ ext{radians} = ext{degrees} \times \frac{\pi}{180} \]Conversely, to convert from radians to degrees, multiply by:\[ ext{degrees} = ext{radians} \times \frac{180}{\pi} \]Understanding both units allows flexibility in solving trigonometric problems, ensuring you choose the appropriate one for the context.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles and sides of triangles to find unknown measurements. These functions are foundational in mathematics, modeling wave patterns, and in the study of periodic phenomena.

Here’s a brief overview:

• Sine ( \( \sin \)): This function returns the ratio of the length of the opposite side to the hypotenuse in a right triangle.
• Cosine ( \( \cos \)): This function returns the ratio of the adjacent side to the hypotenuse.
• Tangent ( \( \tan \)): This function is the ratio of the opposite side to the adjacent side. It can also be expressed as the sine divided by the cosine of an angle.

Trigonometric functions extend beyond triangles and can take any angle as input, expressed in either degrees or radians. They play key roles in analyzing waves, oscillations, and circles, making them essential in both theoretical and applied mathematics.