Problem 54 - Precalculus: A Concise Course

Question

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$ 300^{\circ} $$

Step-by-Step Solution

Answer

The solution is sin(300^\circ) = - \sqrt{3}/2, cos(300^\circ) = 1/2, and tan(300^\circ) = - \sqrt{3}.

Step 1: Convert to Radians

First, convert the angle from degrees to radians. Then, use the relationship \(1 radians = 57.3\, degrees\) to convert 300° into radians. Therefore, \(300^{\circ} = 300 \,/\, 57.3 = 5.24 (approx) radians\)

Step 2: Find equivalent angle in first quadrant

Next, find an equivalent angle in the first quadrant. Since \(300^\circ\) is in the fourth quadrant, the equivalent angle in the first quadrant will be \(360^{\circ} - 300^{\circ} = 60^{\circ}\). This angle can be used to determine the sine and cosine values directly, because they are the coordinates of points on the unit circle. The tangent is derived from the sine and cosine.

Step 3: Determine sine, cosine, and tangent

The values of the sine, cosine, and tangent for angles of \(60^{\circ}\)are known as \(\sin60^{\circ} = \sqrt{3}/2\), \(\cos60^{\circ} = 1/2\), and \(\tan60^{\circ} = \sqrt{3}\). In the fourth quadrant, sine is negative, cosine is positive, and tangent is negative. This yields \(\sin300^{\circ} = -\sqrt{3}/2\), \(\cos300^{\circ} = 1/2\), and \(\tan300^{\circ} = -\sqrt{3}\).

Key Concepts

Sine Function

The sine function

• Denoted as \( \sin \theta \)
• Gives the y-coordinate of the point on the unit circle corresponding to angle \( \theta \)

Understanding the sine of an angle involves imagining a circle with a radius of 1 unit—this is called the unit circle. The angle \( \theta \) represents the angle in question, often originating from the positive x-axis. The sine function is positive in the first and second quadrants and negative in the third and fourth quadrants.
For example, as shown in the original exercise, the sine of \( 300^{\circ} \) is found to be negative because the angle lies in the fourth quadrant. Hence, the sine value becomes \(-\sqrt{3}/2\). So, remember:

• Sine values are aligned with the y-coordinate on the unit circle.
• They change sign based on the angle's quadrant.

Cosine Function

The cosine function

• Denoted as \( \cos \theta \)
• Gives the x-coordinate of the point on the unit circle corresponding to angle \( \theta \)

Cosine is all about the x-axis. The function expresses how far along the x-axis the terminal point of an angle is on the unit circle. It's positive in the first and fourth quadrants and negative in the second and third quadrants.
In the example with \( 300^{\circ} \), we observe that cosine remains positive because \( 300^{\circ} \) belongs to the fourth quadrant, yielding \( 1/2 \). So, key aspects to remember about cosine:

• Cosine expresses proximity along the x-axis.
• It is influenced by the quadrant position, similar to sine.

Tangent Function

The tangent function

• Denoted as \( \tan \theta \)
• Defined as the ratio of the sine to the cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

The tangent function ties together the sine and cosine functions by dividing one by the other. Because of this, the sign of the tangent depends on the signs of both sine and cosine.
If they are both positive or both negative, tangent is positive. If one is positive and the other negative, tangent is negative.

• For \( 300^{\circ} \), sine is negative, and cosine is positive, yielding a negative tangent value, \(-\sqrt{3}\).
• The tangent function helps illustrate how angles interact along both axes at once.

Unit Circle

The unit circle is a powerful concept in understanding trigonometric functions, with a few key points:

• Its radius is always 1.
• Center is at the origin of a coordinate system.

Each point on the unit circle corresponds to \( (\cos \theta, \sin \theta) \), making it easy to find the trigonometric values of angles. The circle is split into four quadrants, each affecting the sign of the trigonometric functions.

• In the first quadrant, both sine and cosine are positive.
• In the second quadrant, sine is positive, cosine negative.
• In the third, both are negative.
• In the fourth, sine is negative, cosine positive.

Understanding the unit circle is fundamental, particularly because it provides visual and tangible ways to assess trigonometric functions for any angle.

Angle Conversion

Angle conversion is essential when working with trigonometric functions because

• Angles can be measured in degrees or radians.

To convert an angle from degrees to radians, the formula used is:\[ \text{Radians} = \left( \text{Degrees} \times \frac{\pi}{180} \right)\]Conversely, to convert from radians to degrees, use:\[ \text{Degrees} = \left( \text{Radians} \times \frac{180}{\pi} \right)\]
This conversion is vital, especially in higher mathematics where radians are often preferable. In the original exercise, converting \( 300^{\circ} \) to radians helps in identifying equivalent angles and utilizing the unit circle efficiently. Understanding both systems allows seamless movement between problems and enhances comprehension of trigonometric identities in broader contexts.