Problem 44 - Precalculus with Calculus Previews

Question

Use a half-angle formula to find the exact value of the given trigonometric function. Do not use a calculator. $$ \sec (-3 \pi / 8) $$

Step-by-Step Solution

Answer

The exact value of \(\sec(-3\pi/8)\) is \(\sqrt{2 + \sqrt{2}}\).

Step 1: Identify the Proper Half-Angle Formula

The secant function, \(\sec(\theta) = \frac{1}{\cos(\theta)}\), requires us to find the cosine of the angle \(-\frac{3\pi}{8}\). To achieve this using a half-angle formula, we rewrite this angle as a half of another angle. The angle \(-\frac{3\pi}{8}\) is half of \(-\frac{3\pi}{4}\) because \(-\frac{3\pi}{4} / 2 = -\frac{3\pi}{8}\).

Step 2: Apply the Half-Angle Formula for Cosine

The half-angle formula for cosine is given by \[\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}\]In our situation, \(\theta = -\frac{3\pi}{4}\). Thus, \[\cos\left(-\frac{3\pi}{8}\right) = \pm \sqrt{\frac{1 + \cos(-\frac{3\pi}{4})}{2}}\]

Step 3: Calculate \(\cos(-\frac{3\pi}{4})\)

The angle \(-\frac{3\pi}{4}\) is in the third quadrant where cosine is negative. The reference angle in the unit circle is \(\frac{\pi}{4}\), so \(\cos\left(-\frac{3\pi}{4}\right) = -\cos\left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}\).

Step 4: Simplify Using the Half-Angle Formula

Substitute \(\cos(-\frac{3\pi}{4})\) back into the half-angle formula:\[\cos\left(-\frac{3\pi}{8}\right) = \pm \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}}\]Simplify inside the square root:\[\cos\left(-\frac{3\pi}{8}\right) = \pm \sqrt{\frac{2/2 - \sqrt{2}/2}{2}}\]\[= \pm \sqrt{\frac{2 - \sqrt{2}}{4}}\]Take the square root of the fraction:\[\cos\left(-\frac{3\pi}{8}\right) = \pm \frac{\sqrt{2 - \sqrt{2}}}{2}\]

Step 5: Determine the Correct Sign

Since \(-\frac{3\pi}{8}\) is in the fourth quadrant where the cosine function is positive, select the positive root.\[\cos\left(-\frac{3\pi}{8}\right) = \frac{\sqrt{2 - \sqrt{2}}}{2}\]

Step 6: Calculate Secant

Now, use the reciprocal identity of secant: \(\sec(\theta) = \frac{1}{\cos(\theta)}\).\[\sec\left(-\frac{3\pi}{8}\right) = \frac{1}{\frac{\sqrt{2 - \sqrt{2}}}{2}} = \frac{2}{\sqrt{2 - \sqrt{2}}}\]

Step 7: Rationalize the Denominator

To eliminate the square root from the denominator, multiply by the conjugate:\[\sec\left(-\frac{3\pi}{8}\right) = \frac{2}{\sqrt{2 - \sqrt{2}}} \times \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2 + \sqrt{2}}} = \frac{2\sqrt{2 + \sqrt{2}}}{(2 - \sqrt{2})(2 + \sqrt{2})}\]Simplify using the difference of squares: \[= \frac{2\sqrt{2 + \sqrt{2}}}{4 - 2} = \frac{2\sqrt{2 + \sqrt{2}}}{2} = \sqrt{2 + \sqrt{2}}\]

Key Concepts

secant function

The secant function, abbreviated as \(\sec(\theta)\), is a trigonometric function that is defined as the reciprocal of the cosine function. Therefore, the formula for secant is \(\sec(\theta) = \frac{1}{\cos(\theta)}\). This means, to evaluate \(\sec(\theta)\) for a specific angle, you first find \(\cos(\theta)\), and then take its reciprocal.

This function is particularly useful when dealing with reciprocal identities. For example, in situations where you have information about the cosine of an angle, you can easily compute the secant by applying this identity. This property comes in handy when solving many trigonometric equations and problems, such as when using the half-angle formulas or dealing with angles on the unit circle.

half-angle formula

The half-angle formulas are extremely useful in trigonometry to find the values of trigonometric functions for half of a given angle. For the cosine function, the half-angle formula is:

• \[\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}\]

To apply this formula correctly, it is important to identify the angle \(\theta\) such that your target angle is exactly half of it. Moreover, a crucial aspect involves choosing the correct sign (\(+\) or \(-\)) of the square root. This depends on the quadrant in which the resulting angle is located.

For instance, if the final angle is in a quadrant where cosine is positive, you select the positive root, as shown in the solution when calculating \(\cos\left(-\frac{3\pi}{8}\right)\). Understanding and correctly applying the half-angle formulas simplifies complex trigonometric problems significantly.

unit circle

The unit circle is a fundamental concept in trigonometry, providing a graphical way to understand angle measures and trigonometric function values. It is a circle with a radius of 1, centered at the origin (0,0) of the coordinate plane. The unit circle allows for easy visualization of angles in radians and the calculation of sine, cosine, and other trigonometric functions.

Angles are measured from the positive x-axis, counterclockwise being positive. Knowing the specific coordinates of common angles, like \(\frac{\pi}{4}, \frac{\pi}{3}, and \pi\), as well as directional sine and cosine signs for each quadrant, is essential when using the unit circle to solve problems. For example, understanding that the cosine of \(\frac{3\pi}{4}\) is negative in the unit circle's third quadrant helps in calculating other angle measures using trigonometric identities.

rationalizing the denominator

Rationalizing the denominator is a mathematical technique used to eliminate square roots or irrational numbers from the bottom of a fraction. In our specific example, we had the expression \(\frac{2}{\sqrt{2 - \sqrt{2}}}\), and to rationalize this denominator, multiplying by the conjugate helps us remove the square root.

To do this:

• Multiply both the numerator and the denominator by the conjugate of the denominator (\(\sqrt{2 + \sqrt{2}}\))

This results in a simplification by taking advantage of the difference of squares, i.e., \((a - b)(a + b) = a^2 - b^2\). Applying this property allows you to replace the denominator with a rational number, in our case reducing to 2, leading to a simplified final result that is easier to interpret and use in further calculations. Rationalizing the denominator is crucial, especially in fields where precise and simplified results are necessary.