Question

a) Arrange the following values of sine in increasing order. \(\sin 1, \sin 2, \sin 3, \sin 4\) b) Show what the four values represent on a diagram of the unit circle. Use your diagram to justify the order from part a). c) Predict the correct increasing order for \(\cos 1, \cos 2, \cos 3,\) and \(\cos 4 .\) Check with a calculator. Was your prediction correct?

Short answer

The increasing order of sine values is \( \text{sin 4}, \text{sin 3}, \text{sin 1}, \text{sin 2} \). For cosine values, the increasing order is \( \text{cos 3}, \text{cos 4}, \text{cos 2}, \text{cos 1} \).

Step-by-step solution

- Find the values of sine

Using a calculator, find the values for \(\text{sin 1}, \text{sin 2}, \text{sin 3}, \text{sin 4}\): - \(\text{sin 1} \approx 0.8415\)- \(\text{sin 2} \approx 0.9093\)- \(\text{sin 3} \approx 0.1411\)- \(\text{sin 4} \approx -0.7568\)

- Arrange the values in increasing order

Order the sine values from lowest to highest: - \(\text{sin 4} \approx -0.7568\)- \(\text{sin 3} \approx 0.1411\)- \(\text{sin 1} \approx 0.8415\)- \(\text{sin 2} \approx 0.9093\)

- Plot the values on the unit circle

Draw a unit circle and mark the angles \(\theta = 1\), \(\theta = 2\), \(\theta = 3\) and \(\theta = 4\) (in radians). Observe the sine values as the y-coordinates of these points.

- Justify the order using the diagram

From the unit circle diagram, observe that as you move from \(\theta = 4\) to \(\theta = 3\), \(\theta = 1\), and \(\theta = 2\), you see the y-coordinates (sine values): \(\text{sin 4} \approx -0.7568\), \(\text{sin 3} \approx 0.1411\), \(\text{sin 1} \approx 0.8415\), \(\text{sin 2} \approx 0.9093\). This confirms the order.

- Predict the order for cosine values

From the unit circle properties, cosine values are the x-coordinates. Predict the order for \(\text{cos 1}, \text{cos 2}, \text{cos 3}, \text{cos 4}\). Typically, these values decrease as the angle moves counterclockwise.

- Verify with calculator

Using a calculator, find the values for \(\text{cos 1}, \text{cos 2}, \text{cos 3}, \text{cos 4}\): - \(\text{cos 1} \approx 0.5403\)- \(\text{cos 2} \approx -0.4161\)- \(\text{cos 3} \approx -0.989992\)- \(\text{cos 4} \approx -0.6536\). The increasing order is \(\text{cos 3} \approx -0.989992\), \(\text{cos 4} \approx -0.6536\), \(\text{cos 2} \approx -0.4161\), \(\text{cos 1} \approx 0.5403\).

Key concepts

Understanding the Sine Function

The sine function is a fundamental concept in trigonometry. It maps an angle, measured in radians, to a value between -1 and 1 on the unit circle. This value represents the y-coordinate of the point on the unit circle at that angle.

Key characteristics of the sine function include:

• The sine of 0 is 0
• The sine of \(\frac{\beta}{2}\) is 1
• The sine of \(\beta\) is 0
• The sine of \(\frac{3\beta}{2}\) is -1

On the unit circle, the sine values are highest at \(\frac{\beta}{2}\) and lowest at \(\frac{3\beta}{2}\).

In our problem, we have angles such as \(1, 2, 3\), and \(4\) radians which all land us in different quadrants.

• For \(1\) radian, the sine value is approximately 0.8415
• For \(2\) radians, the sine value is approximately 0.9093
• For \(3\) radians, the sine value is approximately 0.1411
• For \(4\) radians, the sine value is approximately -0.7568

This helps us arrange the values from lowest to highest as: -0.7568, 0.1411, 0.8415, and 0.9093.

Using the unit circle, it's clear that the y-coordinates of these angles justify our order.

Exploring the Cosine Function

The cosine function is another key trigonometric function. It maps an angle, measured in radians, to a value between -1 and 1 on the unit circle. This value represents the x-coordinate of the point on the unit circle at that angle.

Key characteristics of the cosine function include:

• The cosine of 0 is 1
• The cosine of \(\frac{\beta}{2}\) is 0
• The cosine of \(\beta\) is -1
• The cosine of \(\frac{3\beta}{2}\) is 0

On the unit circle, the cosine values are maximum at 0 radians and minimum at \(\beta\) radians.

In our problem, the angles \(1, 2, 3,\) and \(4\) radians yield the following cosine values:

• For \(1\) radian, the cosine value is approximately 0.5403
• For \(2\) radians, the cosine value is approximately -0.4161
• For \(3\) radians, the cosine value is approximately -0.989992
• For \(4\) radians, the cosine value is approximately -0.6536

When we arrange these values from lowest to highest, we get: -0.989992, -0.6536, -0.4161, and 0.5403.

This ordering aligns with the general behavior of cosine values as the angle moves counterclockwise.

Understanding Radians

Radians are a way of measuring angles. They provide a direct relationship between the angle and the arc length on a unit circle.

One radian is the angle at the center of a circle that subtends an arc length equal to the radius of the circle. There are approximately \(2\beta\) radians in a full circle. Hence, \(1, 2, 3,\), and \(4\) radians refer to angles of varying magnitudes around the circle.

• \(1\) radian is just slightly more than \(57.3\) degrees.
• \(2\) radians is slightly less than 120 degrees.
• \(3\) radians is just under 180 degrees.
• \(4\) radians is over 230 degrees.

Using radians to measure angles is essential in trigonometry as it simplifies many calculations and formulas.

To effectively use the unit circle for solving problems, it's crucial to understand how radians work and how they relate to the circle's coordinates. For instance, as you move counterclockwise from zero radians, both sine and cosine values start at their respective maxima (1 for cosine and 0 for sine) and cycle through their ranges.