Problem 31 - Algebra and Trigonometry with Analytic Geometry

Question

Approximate the acute angle \(\theta\) to the nearest (a) \(0.01^{\circ}\) and (b) \(1^{\prime}\). $$ \sec \theta=4.246 $$

Step-by-Step Solution

Answer

(a) \( 76.34^{\circ} \), (b) \( 76^{\circ} \ 20' \)."

Step 1: Recall the Relationship of Secant and Cosine

The secant of an angle \( \theta \) is the reciprocal of the cosine of that angle. So if \( \sec \theta = 4.246 \), then \( \cos \theta = \frac{1}{4.246} \).

Step 2: Calculate cosine of theta

Using the given \( \sec \theta = 4.246 \), calculate \( \cos \theta \) by finding the reciprocal: \[ \cos \theta = \frac{1}{4.246} \approx 0.2357 \].

Step 3: Use Inverse Cosine to Find Theta

To find \( \theta \), use the inverse cosine function: \( \theta = \cos^{-1}(0.2357) \). This calculation can be done using a calculator that provides trigonometric inverse functions. \( \theta \approx 76.34^{\circ} \).

Step 4: Approximate to the Nearest 0.01 Degree

According to part (a), round \( \theta \approx 76.34^{\circ} \) to two decimal places: \( \theta = 76.34^{\circ} \).

Step 5: Convert to Minutes for Part b

For part (b), convert the decimal degrees to degrees and minutes. Since \( 0.34 \) of a degree equals \( 0.34 \times 60 = 20.4 \) minutes, round 20.4 to the nearest whole number. This gives \( \theta = 76^{\circ} \ 20' \).

Key Concepts

Secant

When exploring trigonometry, understanding the various trigonometric functions and their relationships is essential. One such function is the secant. The secant of an angle in a right triangle is defined as the ratio of the length of the hypotenuse to the length of the adjacent side.

• Mathematically, secant is represented as: \[ \sec \theta = \frac{1}{\cos \theta} \]
• It is important to understand that secant is the reciprocal of cosine. This means secant and cosine functions are deeply connected.
• For example, if \( \sec \theta = 4.246 \), then \( \cos \theta = \frac{1}{4.246} \).

Secant is particularly useful in problems requiring the understanding of the reciprocal trigonometric values, such as what we've done in our given example.

Cosine

Cosine is another fundamental trigonometric function. In a right triangle, it represents the ratio of the adjacent side to the hypotenuse. The cosine function is pivotal not only in geometry but also in calculus and physics.

• Typically noted as \( \cos \theta \), it is connected to the secant function through the reciprocal relationship: \[ \cos \theta = \frac{1}{\sec \theta} \]
• In our exercise, to find \( \cos \theta \) when \( \sec \theta = 4.246 \), we compute:\[ \cos \theta = \frac{1}{4.246} \approx 0.2357 \]

These steps show how cosine values can be calculated from secant values and are crucial in many real-world and theoretical applications.

Inverse Cosine

The inverse cosine function, often denoted as \( \cos^{-1} \), helps to find the angle when the cosine of the angle is known. This function is essential for determining angles, especially in scenarios involving non-integral cosine values.

• To find \( \theta \) from \( \cos \theta \), use the inverse cosine: \[ \theta = \cos^{-1}(\cos \theta) \]
• In our example with \( \cos \theta \approx 0.2357 \), we compute:\[ \theta = \cos^{-1}(0.2357) \approx 76.34^{\circ} \]

Using a calculator is vital for these calculations as they often involve non-standard angles. Understanding inverse cosine is a key part of trigonometry, allowing for determination of angles in various geometrical and real-life problems.