Problem 108 - A Graphical Approach to Precalculus with Limits

Question

Find all values of \(\theta\) if \(\theta\) is in the interval \(\left[0^{\circ}, 360^{\circ}\right)\) function value. Do not use a calculator. $$\sec \theta=-\sqrt{2}$$

Step-by-Step Solution

Answer

\(\theta = 135^{\circ}\) and \(\theta = 225^{\circ}\) are the solutions.

Step 1 - Recall Trigonometric Identities

Recall that \(\sec \theta = \frac{1}{\cos \theta}\). Hence, if \(\sec \theta = -\sqrt{2}\), then \(\frac{1}{\cos \theta} = -\sqrt{2}\).

Step 2 - Solve for \(\cos \theta\)

Transform the equation \(\frac{1}{\cos \theta} = -\sqrt{2}\) to solve for \(\cos \theta\). This gives \(\cos \theta = -\frac{1}{\sqrt{2}}\). Rationalize the denominator to get \(\cos \theta = -\frac{\sqrt{2}}{2}\).

Step 3 - Determine the General Angle

Recall that \(\cos \theta = -\frac{\sqrt{2}}{2}\) at \(\theta = 135^{\circ}\) and \(\theta = 225^{\circ}\). These angles correspond to where the reference angle is \(45^{\circ}\) but \(\cos\) is negative in the 2nd (Q2) and 3rd (Q3) quadrants.

Step 4 - Check for Incorrect Answers

Ensure none of the angles provided go beyond the interval \([0^{\circ}, 360^{\circ})\). Both \(135^{\circ}\) and \(225^{\circ}\) lie within this interval.

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is one of the six fundamental trigonometric functions. In essence, secant is the reciprocal of the cosine function. This means that \( \sec \theta = \frac{1}{\cos \theta} \). Understanding this reciprocal relationship is crucial when solving trigonometric equations involving the secant function.
To find the value of \( \sec \theta = -\sqrt{2} \), as given in the exercise, you need to comprehend what this statement represents. It implies that the cosine of the angle \( \theta \) is the reciprocal of \(-\sqrt{2}\), which needs further simplification. This is where converting secant back to its cosine equivalent becomes valuable. Once you have identified the cosine value, you can then work within the familiar realm of cosine angles.

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables involved. These identities are fundamental tools in trigonometry, providing connections between functions that can simplify complex equations. For example, knowing that \( \sec \theta = \frac{1}{\cos \theta} \) enables you to transform an equation involving secant into one that involves cosine, which is often easier to solve.
Utilize these identities to find relationships between angles and their trigonometric values:

• The Pythagorean identities, such as \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Reciprocal identities, like \( \csc \theta = \frac{1}{\sin \theta} \).
• Co-function identities that include \( \sin(90^\circ - \theta) = \cos \theta \).

By using these identities, you can simplify expressions and solve problems effectively without heavy reliance on calculators.

Cosine Function

The cosine function, written as \( \cos \theta \), is a key player in trigonometry. It measures the horizontal coordinate of a point on a unit circle. Recognizing cosine values and their corresponding angles is essential for solving trigonometric problems.
The key task in the exercise is to find values where \( \cos \theta = -\frac{\sqrt{2}}{2} \). These values occur at angles where the reference angle is \( 45^\circ \), but cosines take negative values in the quadrants 2 and 3. In the standard unit circle:

• \(135^\circ\) is in the second quadrant.
• \(225^\circ\) is in the third quadrant.

This understanding will help you locate the exact angles required. Always ensure these angles fall within the given interval to prevent errors.