Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-12,\) find the radian measure of each angle whose degree measure is given. \(90^{\circ}\)

Step-by-Step Solution

Answer

The radian measure of \(90^\circ\) is \(\frac{\pi}{2}\).

Step 1: Understanding the Problem

We need to convert an angle given in degrees, specifically \(90^\circ\), into radians. The conversion formula from degrees to radians is crucial to solve this problem.

Step 2: Formula for Conversion

The standard conversion formula from degrees to radians is \( \theta \, \text{radians} = \frac{\theta^\circ \times \pi}{180} \).

Step 3: Substitute the Degree Value

We substitute \(90\) for \(\theta^\circ\) in the formula: \[ \frac{90 \times \pi}{180} \].

Step 4: Simplify the Fraction

Simplify the fraction \( \frac{90}{180} \). This simplifies to \( \frac{1}{2} \).

Step 5: Final Calculation

Now, multiply the simplified fraction by \(\pi\):\[ \frac{1}{2} \times \pi = \frac{\pi}{2} \].

Step 6: Conclusion

Therefore, the radian measure of \(90^\circ\) is \( \frac{\pi}{2} \).

Key Concepts

Radian Measure

The concept of radian measure is fundamental in understanding angles in trigonometry. While degrees are more commonly used in everyday contexts, radians provide a more natural way of describing angles in mathematical calculations. A radian is the angle at the center of a circle that subtends an arc equal in length to the radius of the circle. This might sound a bit complicated, but it simply means using the circle's own dimensions to measure angles. There are key properties of radians you should remember:

• One full circle is equal to 2π radians, since a circle's circumference (dependent on its radius) is 2π.
• Half a circle (180°) is π radians, because dividing a circle in half cuts its circumference to 1π.
• Quarter circle or right angle (90°) equates to π/2 radians, as 90° is half of 180°.

Radians make calculations involving angular velocity and trigonometry much more straightforward.

Angle Conversion

Converting an angle from degrees to radians is an essential skill in trigonometry. This conversion process is straightforward if you remember the relationship: Since a circle is 360° in degrees and 2π in radians, we use the ratio of these to convert degrees to radians. The conversion formula is:\[ \theta \, \text{radians} = \frac{\theta^\circ \times \pi}{180} \]Illustrating with an example, to convert 90° to radians:

• We apply the formula: \( \frac{90 \times \pi}{180} \)
• Simplify the fraction: \( \frac{90}{180} = \frac{1}{2} \)
• Thus, \( 90^\circ = \frac{\pi}{2} \text{ radians} \)

Remembering this ratio and conversion method will help you easily switch between these two units of angular measurement.

Trigonometry

Trigonometry is the branch of mathematics dealing with angles and the relationships between the sides of triangles. Understanding how different angle measures, like radians and degrees, function is crucial in trigonometry.Some foundational trigonometric concepts:

• Sine, cosine, and tangent functions relate the angles to the ratios of the sides of a right triangle.
• Trigonometric identities, such as the Pythagorean identity, are useful in proving equations: \( \sin^2\theta + \cos^2\theta = 1 \).
• The unit circle is a powerful tool in understanding trigonometric functions, as it displays angles measured in radians around a circle with a radius of one.

Trigonometry commonly uses radians because they simplify many calculations, making them more efficient for solving problems involving periodic functions and circular motion.