Problem 4 - Algebra 2

Question

Find the value of each expression. $$ \sec \theta, \text { if } \tan \theta=-1 ; 270^{\circ}<\theta<360^{\circ} $$

Step-by-Step Solution

Answer

\(\sec \theta = \sqrt{2}\) for \(\theta = 315^{\circ}\).

Step 1: Determine the Quadrant

Since we have \(270^{\circ} < \theta < 360^{\circ}\), the angle \(\theta\) is located in the fourth quadrant. The fourth quadrant is where sine is negative, cosine is positive, and tangent is negative.

Step 2: Understand the given condition

We are given \(\tan \theta = -1\). This means the tangent of the angle has the magnitude of 1 and is negative in the fourth quadrant.

Step 3: Recall the exact tangent value

The exact value of \(\tan \theta = -1\) occurs when \(\theta = 315^{\circ}\). Therefore, \(\theta = 315^{\circ}\) since it is the angle that satisfies both the given tangent value and the quadrant.

Step 4: Find the secant function value

We know that \(\sec \theta = \frac{1}{\cos \theta}\). For \(\theta = 315^{\circ}\), \(\cos 315^{\circ} = \frac{1}{\sqrt{2}}\). So, \(\sec 315^{\circ} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2}\).

Key Concepts

Secant Function

The secant function is a trigonometric function represented by \( \sec \theta = \frac{1}{\cos \theta} \). It is the reciprocal of the cosine function. Reciprocals play a crucial role in trigonometry and can help in finding values when cosine is known.
Understanding secant is vital as it helps solve problems involving right triangles and is often used in calculus. The secant function increases complexity in trigonometric problems but remains simple if the cosine value is known.

• The secant function is undefined where cosine is zero, as division by zero is undefined.
• The graph of the secant function has vertical asymptotes where \( \cos \theta = 0 \).
• Secant provides values for angles that complement sine and cosine relationships.

Mastering the secant function enriches problem-solving skills in mathematics.

Tangent Function

The tangent function, denoted as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), is another fundamental trigonometric function. It represents the ratio of the sine to the cosine of an angle. The value of the tangent function is crucial in determining angles and calculations involving right triangles.
In this exercise, we are given \( \tan \theta = -1 \). This indicates that the sine and cosine must be equal in magnitude but opposite in sign. In the fourth quadrant, this scenario means one is positive and the other is negative. Here, cosine is positive, and sine is negative, hence \( \tan \theta = \frac{-1}{1} = -1 \).

• The value \( -1 \) for tangent typically corresponds to an angle of \( 315^{\circ} \) or \( 7\pi/4 \), in the fourth quadrant.
• Tangent helps in understanding the slope of angles and the steepness of curves in graphical representations.

The tangent has applications in calculating distances and angles in various fields such as physics and engineering.

Fourth Quadrant

The fourth quadrant in the coordinate system refers to angles ranging from \( 270^{\circ} \) to \( 360^{\circ} \). In this region, different trigonometric functions take on specific values:

• Sine is negative.
• Cosine is positive.
• Tangent is negative.

Understanding the sign changes in this quadrant is important:
For instance, the provided problem indicates the angle is in the fourth quadrant where tangent is negative. This aligns with the identity \( \tan \theta = \frac{\text{negative}}{\text{positive}} = -1 \).
Knowing which trigonometric functions hold positive or negative values aids in solving various geometric and trigonometric problems.

Exact Angle Values

Exact angle values are specific angles at which the trigonometric functions take on well-known values. It is important to memorize key exact angles as they simplify many trigonometric calculations. For example, angles like \( 0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}, \) and \( 90^{\circ} \) are often studied for their precise sine, cosine, and tangent values.

• \( \tan \theta = -1 \) corresponds to \( 315^{\circ} \) in the fourth quadrant, as seen in the current problem.
• \( 315^{\circ} \) aligns with the coordinate \( \chi = (\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}) \) where \( \,\cos 315^{\circ} = \frac{1}{\sqrt{2}} \,\text{and}\tan 315^{\circ} = -1 \).

Having such exact values at hand provides a powerful tool for both theoretical and applied mathematics. Recognizing these angles helps decipher more complex trigonometric expressions.