Problem 66 - A Graphical Approach to Precalculus with Limits

Question

Find exact values of the six trigonometric functions for each angle by hand. Do not use a calculator. $$420^{\circ}$$

Step-by-Step Solution

Answer

The six trigonometric functions for \(420^{\circ}\) are: \(\sin = \frac{\sqrt{3}}{2}\), \(\cos = \frac{1}{2}\), \(\tan = \sqrt{3}\), \(\csc = \frac{2\sqrt{3}}{3}\), \(\sec = 2\), \(\cot = \frac{\sqrt{3}}{3}\).

Step 1: Determine the Reference Angle

First, we need to find the reference angle for \(420^{\circ}\). Since a full circle is \(360^{\circ}\), we can subtract \(360^{\circ}\) from \(420^{\circ}\) to get the reference angle. \(420^{\circ} - 360^{\circ} = 60^{\circ}\). Thus, the reference angle is \(60^{\circ}\).

Step 2: Identify the Quadrant

Since \(420^{\circ}\) is equivalent to \(60^{\circ}\), we need to determine in which quadrant \(420^{\circ}\) lies. To do this, observe that \(420^{\circ}\) can be divided into a full rotation of \(360^{\circ}\) plus an additional \(60^{\circ}\), placing it in the first quadrant.

Step 3: Find Trigonometric Values for \(60^{ ext{o}}\)

In the first quadrant, the trigonometric values for \(60^{\circ}\) are:- \( \sin(60^{\circ}) = \frac{\sqrt{3}}{2} \)- \( \cos(60^{\circ}) = \frac{1}{2} \)- \( \tan(60^{\circ}) = \sqrt{3} \)Since \(420^{\circ}\) is in the first quadrant and equivalent to \(60^{\circ}\), the values are the same for sine, cosine, and tangent.

Step 4: Derive Reciprocal Functions

Now we find the reciprocal trigonometric functions:- \( \csc(60^{\circ}) = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \)- \( \sec(60^{\circ}) = 2 \)- \( \cot(60^{\circ}) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \)

Key Concepts

Reference Angle

When considering angles beyond one full turn of a circle, like our example angle of \(420^{\circ}\), the reference angle becomes very useful. The reference angle is essentially the smallest angle that the given angle shares with the x-axis. This can always be found by:

• Subtracting \(360^{\circ}\) from your angle until you are in the range of \(0^{\circ} \leq \text{angle} \leq 360^{\circ}\).
• In the case of \(420^{\circ}\), we subtract \(360^{\circ}\) once and end up with \(60^{\circ}\).

The reference angle is thus \(60^{\circ}\), which means the key trigonometric values associated with \(60^{\circ}\) will be helpful when evaluating \(420^{\circ}\). Remember, the reference angle provides us with the framework to determine all trigonometric values for any angle, without needing to use a calculator.

Quadrants

The 'quadrant' refers to one of the four sections of the coordinate plane in which an angle can terminate. When analyzing an angle, knowing its quadrant can tell us the signs (positive or negative) of its trigonometric functions. Here's how:

• The first quadrant (\(0^{\circ}\) to \(90^{\circ}\)): all trigonometric functions are positive here.
• The second quadrant (\(90^{\circ}\) to \(180^{\circ}\)): sine is positive, cosine and tangent are negative.
• The third quadrant (\(180^{\circ}\) to \(270^{\circ}\)): tangent is positive, sine and cosine are negative.
• The fourth quadrant (\(270^{\circ}\) to \(360^{\circ}\)): cosine is positive, sine and tangent are negative.

For \(420^{\circ}\), which translates to \(60^{\circ}\) in the first quadrant, all trigonometric functions take their positive values. This knowledge simplifies understanding the behavior of angles beyond \(360^{\circ}\).

Reciprocal Trigonometric Functions

In addition to the primary trigonometric functions, we have their reciprocals: cosecant, secant, and cotangent. Understanding these is crucial for solving more complex trigonometry problems.

• Cosecant \((\csc)\) is the reciprocal of sine: \( \csc(\theta) = \frac{1}{\sin(\theta)} \).
• Secant \((\sec)\) is the reciprocal of cosine: \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
• Cotangent \((\cot)\) is the reciprocal of tangent: \( \cot(\theta) = \frac{1}{\tan(\theta)} \).

For our angle \(60^{\circ}\), which shares its trigonometric values with \(420^{\circ}\) since both are effectively in the same quadrant, these reciprocals will be located as follows:

• \(\csc(60^{\circ}) = \frac{2\sqrt{3}}{3}\)
• \(\sec(60^{\circ}) = 2\)
• \(\cot(60^{\circ}) = \frac{\sqrt{3}}{3}\)

By understanding reciprocal trigonometric functions, we can handle trigonometric equations more effectively and tackle a wide array of real-world applications.