Problem 12 - Algebra and Trigonometry with Analytic Geometry

Question

Find the exact radian measure of the angle. (a) \(630^{\circ}\) (b) \(54^{\circ}\) (c) \(95^{\circ}\)

Step-by-Step Solution

Answer

(a) \( \frac{7\pi}{2} \) radians, (b) \( \frac{3\pi}{10} \) radians, (c) \( \frac{19\pi}{36} \) radians.

Step 1: Convert 630° to Radians

We use the conversion formula from degrees to radians, which is \( \pi \text{ radians} = 180^{\circ} \). Therefore, the conversion factor is \( \frac{\pi}{180} \). Multiply this factor by 630° to find the radian measure: \( 630^{\circ} \times \frac{\pi}{180} = \frac{630\pi}{180} \). Simplifying, we get \( \frac{21\pi}{6} = \frac{7\pi}{2} \) radians.

Step 2: Convert 54° to Radians

Using the conversion formula again, multiply 54° by \( \frac{\pi}{180} \): \( 54^{\circ} \times \frac{\pi}{180} = \frac{54\pi}{180} \). Simplify the fraction to get \( \frac{3\pi}{10} \) radians.

Step 3: Convert 95° to Radians

Multiply 95° by the conversion factor \( \frac{\pi}{180} \) to get \( 95^{\circ} \times \frac{\pi}{180} = \frac{95\pi}{180} \). Simplify the fraction to \( \frac{19\pi}{36} \) radians.

Key Concepts

Degree to Radian Conversion

Converting degrees to radians is a fundamental skill in trigonometry and angle measurement. This transformation is crucial when working with angles in various mathematical and real-world applications where radians are often the preferred unit of measure over degrees. To convert degrees into radians, you use the following standard formula:\[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \]This formula arises from the fact that a full circle is composed of 360 degrees, which is equivalent to \(2\pi\) radians. Therefore, each degree is equivalent to \(\frac{\pi}{180}\) radians. For example, using this formula, a conversion from 630 degrees results in:- \( 630^{\circ} \times \frac{\pi}{180} = \frac{7\pi}{2} \) radians.It's important to simplify the resulting fraction as much as possible to make it more convenient for calculations and further use.

Angle Measurement

Understanding angle measurement is essential in both mathematics and various scientific fields. Angles can be measured in two primary units: degrees and radians. Each has unique uses depending on the context.- **Degrees:** Commonly used in everyday applications, dividing a circle into 360 equal parts. - **Radians:** Frequently used in mathematics and physics, particularly in advanced topics like calculus and trigonometric functions.Radians measure the angle by the length of an arc formed by the angle on the circumference of a circle. One radian is the angle formed when the radius cuts off an arc equal in length to the radius itself.Switching from degrees to radians also facilitates certain calculations in trigonometry by aligning more directly with mathematical constants like \(\pi\), thus simplifying expressions and computations.

Trigonometry

Trigonometry is the branch of mathematics focused on the study of angles, triangles, and trigonometric functions. Understanding radians is crucial in trigonometry as it facilitates the study of circular and oscillatory motions, among other phenomena. - **Trigonometric Functions:** Employ radians to provide smooth, continuous functions. For instance, the sine, cosine, and tangent functions are periodic and express well when angles are in radian measure. - **Applications:** Extend beyond mathematics into physics, engineering, and other sciences. Trigonometry principles govern problems involving waves, oscillations, and periodic phenomena. Radian measures tend to simplify computations in calculus and other mathematical applications due to their natural relation to the properties of circles and periodic functions. For students, mastering radian conversion and its application in trigonometry is crucial for advanced mathematical studies.