Problem 69 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the reference angle \(\theta^{\prime} .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\boldsymbol{\theta}=1.8$$

Step-by-Step Solution

Answer

The reference angle, \(\theta^{\prime}\), is equal to the given angle, \(\theta\), when \(\theta\) is 1.8. Therefore, \(\theta^{\prime} = 1.8\).

Step 1: Compare the input angle with \(\pi\)

To determine the reference angle, look at the given angle, which is \(1.8\) radians. This angle is positive and less than \(\pi\). Therefore, its reference angle is the same as the given angle.

Step 2: Sketch \(\theta\) in standard position and label \(\theta^{\prime}\)

To visualize this, you can draw a unit circle. By marking the angle \(1.8\) radians with a radius, you locate it in what quadrant it is. Here, because it's less than \(\pi\), it falls in the second quadrant. The projection onto the x-axis always yields an angle that is positive and <= \(\pi/2\), here we simply get the original angle as it's less than \(\pi\) and positive. Therefore \(\theta^{\prime} = 1.8\)

Key Concepts

Angles in Radians

When we measure angles, an important unit we can use is called radians. Unlike degrees, which break a circle into 360 parts, radians relate directly to the circle's radius. One full circle in radians is equal to the circumference of the circle divided by the radius, which is exactly \(2\pi\) radians. Thus, half a circle is \(\pi\) radians, a quarter is \(\pi/2\) radians, and so on.

Using radians makes calculations with circles and angles more natural. For example, to find angles or arc lengths, formulas often become simpler with radians. This is why they are frequently used in mathematics and physics.

In our exercise, the angle is given as \(1.8\) radians, which is slightly more than half of \(\pi/2 = 1.57\). It is important to know if the angle is more or less than \(\pi\) because that determines its reference angle.

Standard Position

An angle is said to be in "standard position" when its vertex is at the origin of a coordinate plane, its initial side lies along the positive x-axis, and it rotates in the counterclockwise direction to its terminal side.

This convention helps in understanding and identifying the position of an angle easily. If an angle is positive, like \(1.8\) radians in our exercise, it will rotate counterclockwise from the positive x-axis. Negative angles rotate clockwise.

In the exercise, placing our \(1.8\) radians angle in standard position allows us to see that it lies in the second quadrant. This is because any angle greater than \(\pi/2\) (about 1.57 radians) but less than \(\pi\) (about 3.14 radians) places it there. Standard position helps us sketch angles and find their reference angles.

Unit Circle

The unit circle is a critical concept in trigonometry. It's a circle with a radius of one, centered at the origin of a coordinate plane.

On the unit circle, every point has coordinates \( (\cos(\theta), \sin(\theta))\), making it a powerful tool to understand trigonometric functions. The x-coordinate represents the cosine of the angle, while the y-coordinate represents its sine, thus linking angle measures with coordinate geometry.

In the exercise, when you draw the unit circle and plot the angle \(1.8\) radians, you'll be visualizing how it passes through certain coordinates based on its sine and cosine values, which can help you confirm the quadrant.

The unit circle also makes finding the reference angle straightforward. The reference angle is always the smallest angle the terminal side makes with the x-axis, thus helping to simplify trigonometry problems.