Problem 31 - Algebra 2

Question

Find the value of each expression. \(\sec \theta,\) if \(\sin \theta=\frac{3}{4} ; 90^{\circ}<\theta<180^{\circ}\)

Step-by-Step Solution

Answer

\(-\frac{4\sqrt{7}}{7}\)

Step 1: Understanding Trigonometric Identities

We are asked to find \(\sec \theta\). The secant function is the reciprocal of the cosine function, i.e., \(\sec \theta = \frac{1}{\cos \theta}\). Given \(\sin \theta = \frac{3}{4}\), we need to find \(\cos \theta\) to get \(\sec \theta\).

Step 2: Applying the Pythagorean Identity

The Pythagorean identity states \(\sin^2 \theta + \cos^2 \theta = 1\). Substituting the given \(\sin \theta = \frac{3}{4}\), we calculate: \(\left(\frac{3}{4}\right)^2 + \cos^2 \theta = 1\). Simplifying gives \(\frac{9}{16} + \cos^2 \theta = 1\).

Step 3: Solving for \(\cos^2 \theta\)

From the equation \(\frac{9}{16} + \cos^2 \theta = 1\), subtract \(\frac{9}{16}\) from both sides to find \(\cos^2 \theta\). This results in \( \cos^2 \theta = 1 - \frac{9}{16}\). Simplifying gives \(\cos^2 \theta = \frac{7}{16}\).

Step 4: Determining \(\cos \theta\)

Taking the square root of both sides, \(\cos \theta = \pm \frac{\sqrt{7}}{4}\). Since \(90^\circ<\theta <180^\circ\), \(\cos \theta\) is negative because it's in the second quadrant. Thus, \(\cos \theta = -\frac{\sqrt{7}}{4}\).

Step 5: Calculating \(\sec \theta\)

Now that we have \(\cos \theta = -\frac{\sqrt{7}}{4}\), we find \(\sec \theta = \frac{1}{\cos \theta}\). Therefore, \(\sec \theta = \frac{1}{-\frac{\sqrt{7}}{4}} = -\frac{4}{\sqrt{7}}\). Rationalizing the denominator gives \(\sec \theta = -\frac{4\sqrt{7}}{7}\).

Key Concepts

Secant Function

The secant function, denoted by \( \sec \theta \), is one of the fundamental trigonometric functions, closely related to the cosine function. In fact, it is the reciprocal of the cosine function. This means that \( \sec \theta = \frac{1}{\cos \theta} \).

• This definition implies that if we know \( \cos \theta \), we can easily determine \( \sec \theta \).
• The function is undefined wherever the cosine function equals zero, because division by zero is undefined.

To solve problems involving the secant function, such as finding its value given other trigonometric information, understanding and using its reciprocal relationship to cosine is crucial. This reciprocal relationship also assists in manipulating and solving various trigonometric equations easily. Understanding these foundational concepts paves the way for tackling complex trigonometric identities and problems.

Pythagorean Identity

The Pythagorean identity is a key concept in trigonometry, establishing a relationship between the sine and cosine functions. It is expressed as:\[ \sin^2 \theta + \cos^2 \theta = 1 \]

• This identity holds for any angle \( \theta \) and is derived from the Pythagorean theorem applied to a unit circle.
• The identity is particularly useful for finding one trigonometric function when the other is known, which is exactly what we did in the original solution.

By knowing \( \sin \theta \) and utilizing the Pythagorean identity, we can solve for \( \cos^2 \theta \). This concept forms the backbone of many trigonometric solutions, allowing easy conversion between sine, cosine, and other related functions. It is a powerful tool because it highlights the intrinsic relationship between these functions.

Sine and Cosine Functions

Sine and cosine functions are fundamental trigonometric functions used to describe relationships in right-angled triangles and unit circles.

• The sine function, \( \sin \theta \), gives the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.
• The cosine function, \( \cos \theta \), provides the ratio of the adjacent side to the hypotenuse.
• When applied to the unit circle, these functions help to define the coordinates of a point on the circle.

Both functions have values ranging from -1 to 1, and they exhibit periodic behavior with a period of \(2\pi\). These properties are extensively used in various real-world applications, including physics, engineering, and computer graphics, to model wave patterns and periodic phenomena. Understanding these basic trigonometric definitions is critical in solving a wide range of mathematical problems.

Trigonometric Functions in Different Quadrants

Understanding how trigonometric functions behave in different quadrants is crucial for correctly determining their signs and values.

• In the first quadrant, all trigonometric functions are positive.
• In the second quadrant, sine is positive while cosine and secant are negative. This relates directly to the problem when \(90^{\circ}<\theta<180^{\circ}\).
• In the third quadrant, tangent and cotangent are positive, while sine and cosine are negative.
• In the fourth quadrant, cosine and secant are positive, while sine is negative.

Knowing the sign changes in different quadrants helps correctly interpret trigonometric values based on the angle's position. In the original exercise, being aware that \( \cos \theta \) is negative in the second quadrant was essential to finding the correct value of \( \sec \theta \). This quadrant system provides a clear structure for recalling trigonometric function behaviors based on angle measurements.