Problem 62 - Precalculus

Question

Evaluate the following expressions exactly: $$\sec 135^{\circ}$$

Step-by-Step Solution

Answer

\(\sec 135^{\circ} = -\sqrt{2}\)

Step 1: Convert to Radians

Convert the angle from degrees to radians because trigonometric functions are often easier to work with in radians. The conversion factor from degrees to radians is \(\frac{\pi}{180}\). So, \(135^{\circ} = 135 \times \frac{\pi}{180} = \frac{3\pi}{4}\) radians.

Step 2: Identify the Reference Angle

Identify the reference angle for \(\frac{3\pi}{4}\). The angle \(\frac{3\pi}{4}\) is in the second quadrant of the unit circle, and the reference angle is \(\pi - \frac{3\pi}{4} = \frac{\pi}{4}\).

Step 3: Calculate the Cosine using the Reference Angle

Since the secant function is the reciprocal of cosine, begin by finding the cosine. The cosine of \(\frac{\pi}{4}\) is \(\frac{1}{\sqrt{2}}\). In the second quadrant, the cosine is negative, so \(\cos \frac{3\pi}{4} = -\frac{1}{\sqrt{2}}\).

Step 4: Find the Secant

The secant is the reciprocal of the cosine. Therefore, \(\sec 135^{\circ} = \sec \frac{3\pi}{4} = \frac{1}{-\frac{1}{\sqrt{2}}} = -\sqrt{2}\).

Step 5: Simplify

The reciprocal \(-\frac{1}{\sqrt{2}}\) simplifies to \(-\sqrt{2}\) since multiplying both numerator and denominator by \(\sqrt{2}\) rationalizes the denominator. Hence, \(\sec 135^{\circ} = -\sqrt{2}\).

Key Concepts

Radians Conversion

Radians are a crucial part of trigonometry, especially when dealing with trigonometric functions. They provide a more natural approach to angle measurement than degrees, especially when combining angles with mathematical operations. A complete circle is represented by \(2\pi\) radians, which is equivalent to 360°.

For conversion, the formula is simple: multiply the angle in degrees by \(\frac{\pi}{180}\). This factor helps transition from a degree measurement to its corresponding radian measure. Take the angle given in the original exercise, \(135^{\circ}\), for example. When we convert this angle by multiplying it by \(\frac{\pi}{180}\), the resulting measure is \(\frac{3\pi}{4}\) radians.

Using radians helps align with many mathematical expressions and formulas used in calculus and higher mathematics.

• A full circle in radian measure is \(2\pi\).
• Radians often simplify equations and functions related to periodicity.
• Converting degrees to radians involves multiplying by \(\frac{\pi}{180}\).

Reference Angle

A reference angle helps simplify the calculation of trigonometric functions by relating the angle to the first quadrant. It essentially measures how far an angle is from the x-axis on the unit circle.

When dealing with angles in radians, such as \(\frac{3\pi}{4}\), it’s crucial to find the reference angle for evaluating trigonometric values. This is often done by determining the angle's position in a specific quadrant and then subtracting from \(\pi\) or \(2\pi\) accordingly.

For \(\frac{3\pi}{4}\), the angle resides in the second quadrant, which means we subtract it from \(\pi\): \[\pi - \frac{3\pi}{4} = \frac{\pi}{4}\].This reference angle \(\frac{\pi}{4}\) is significant because it corresponds to known values of trigonometric functions.

• Reference angles are always between 0 and \(\frac{\pi}{2}\) radians.
• They help find trigonometric functions' values in other quadrants.
• Using reference angles simplifies working with non-first quadrant angles.

Cosine

Cosine is defined as the ratio of the adjacent side to the hypotenuse in a right triangle. However, in the context of the unit circle, it represents the x-coordinate of a point at a given angle.

When determining the cosine of a reference angle such as \(\frac{\pi}{4}\), it is useful to remember that the known value is \(\frac{1}{\sqrt{2}}\) or rationalized as \(\frac{\sqrt{2}}{2}\). In the unit circle, cosine values are positive in the first and fourth quadrants.

In the exercise, you need to find the cosine for \(\frac{3\pi}{4}\). Located in the second quadrant, cosine values are negative. Thus, the cosine for \(\frac{3\pi}{4}\) becomes -\(\frac{1}{\sqrt{2}}\)or -\(\frac{\sqrt{2}}{2}\).

• Cosine is positive in the first and fourth quadrants.
• Use reference angles to determine cosine values in other quadrants.
• The cosine of \(\frac{\pi}{4}\) is \(\frac{1}{\sqrt{2}}\), negative in the second and third quadrants.

Reciprocal Function

The reciprocal of the cosine function is called the secant. In fact, all the primary trigonometric functions have reciprocals: sine to cosecant, cosine to secant, and tangent to cotangent.

Finding the secant involves taking the reciprocal of the cosine value.In the case of our problem, since \(\cos 135^{\circ} = -\frac{1}{\sqrt{2}}\), the secant function becomes the reciprocal of this: \(\frac{1}{-\frac{1}{\sqrt{2}}}\), which simplifies to -\(\sqrt{2}\) when the denominator is rationalized.

When working with reciprocal functions, it's crucial to remember:

• Secant and cosine are reciprocals.
• Multiply by the conjugate to rationalize denominators when recalculate.
• Quadrant matters: secant inherits the sign of cosine depending on the quadrant.