Question

In Exercises 25-32, evaluate the function without using a calculator. \(\sec 135^{\circ}\)

Short answer

-√2

Step-by-step solution

Convert degrees to radians

In order to work with trigonometric functions, it is easier to work in radians than in degrees. Convert 135 degrees into radians. The conversion is done by multiplying the degree by \(\pi /180\). So it will be \( \frac{135\pi}{180}\) which simplifies to \( \frac{3\pi}{4}\)

Determine the cosine of the angle

The cosine of 135 degrees (or \(\frac{3\pi}{4}\) radians) is -√2/2. This cosine value is well-known and can be determined by inspecting the unit circle or using the Pythagorean theorem.

Compute the secant

The secant function is the reciprocal of the cosine function, that is \(\sec \theta = 1/ cos \theta\). Plug the value from Step 2 into this formula to determine the secant at 135 degrees, meaning \(\sec 135^{\circ} = 1/(-√2/2) = -2/√2\). Simplify the expression by multiplying the numerator and denominator by √2 to get -√2.

Key concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry that relates angles to trigonometric functions. It is a circle with a radius of 1, centered at the origin of a coordinate plane. Each point on the unit circle corresponds to an angle and can be used to determine the sine, cosine, and tangent of that angle.

• Cosine values are represented by the x-coordinate of points on the unit circle.
• Sine values are represented by the y-coordinate of these points.
• Tangent can be determined by the ratio of sine to cosine, i.e., y/x.

For an angle of 135 degrees, or \(\frac{3\pi}{4}\) radians, the corresponding point on the unit circle has coordinates \(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\). Consequently, this provides the cosine value needed for evaluating the secant function. This simple yet powerful circle helps us visualize and calculate trigonometric values easily.

Radian Conversion

Radian conversion is an essential process in trigonometry, especially when dealing with angles. Radians offer a natural way to describe angles by illustrating the length of an arc divided by the radius of a circle.

• 1 radian is equivalent to the angle made by taking the radius and wrapping it around the circle’s circumference.
• There are \(2\pi\) radians in a full circle (360 degrees).

To convert degrees to radians, multiply the degree value by \(\frac{\pi}{180}\). For example, to convert 135 degrees, the calculation is \(\frac{135\pi}{180} = \frac{3\pi}{4}\). Radian conversion helps trigonometric calculations align more naturally with the mathematical properties of circles, especially when using calculus.

Reciprocal Function

The reciprocal function in trigonometry is crucial for understanding other trigonometric functions such as secant, cosecant, and cotangent.

• Secant (\(\sec\)) is the reciprocal of cosine (\(\cos\)), meaning \(\sec \theta = \frac{1}{\cos \theta}\).
• Cosecant (\(\csc\)) is the reciprocal of sine (\(\sin\)), or \(\csc \theta = \frac{1}{\sin \theta}\).
• Cotangent (\(\cot\)) is the reciprocal of tangent (\(\tan\)), which translates to \(\cot \theta = \frac{1}{\tan \theta}\).

When evaluating \(\sec 135^{\circ}\), we first calculate its cosine from the unit circle, which is \(-\frac{\sqrt{2}}{2}\). Applying the reciprocal formula gives \(\sec 135^{\circ} = \frac{1}{-\frac{\sqrt{2}}{2}}\), which simplifies to \(-\sqrt{2}\). Reciprocal functions are especially handy when dealing with trigonometric identities and solving equations.