Problem 35 - Algebra and Trigonometry with Analytic Geometry

Question

Approximate, to the nearest \(0.1^{\circ}\), all angles \(\theta\) in the interval \(\left[0^{\circ}, 360^{\circ}\right)\) that satisfy the equation. (a) \(\sin \theta=-0.5640\) (b) \(\cos \theta=0.7490\) (c) \(\tan \theta=2.798\) (d) \(\cot \theta=-0.9601\) (e) \(\sec \theta=-1.116\) (f) \(\csc \theta=1.485\)

Step-by-Step Solution

Answer

(a) 214.4°, 325.6°; (b) 41.8°, 318.2°; (c) 70.9°, 250.9°; (d) 133.9°, 313.9°; (e) 153.5°, 206.5°; (f) 42.3°, 137.7°.

Step 1: Understanding the Problem

We need to approximate angles \( \theta \) within the range of \( [0^{\circ}, 360^{\circ}) \) that satisfy each given trigonometric equation.

Step 2: Solving for \( \sin \theta = -0.5640 \)

1. Use a calculator to find the reference angle using \( \theta = \arcsin(0.5640) \), which gives approximately \( \theta = 34.4^{\circ} \).2. Since sin is negative in the third and fourth quadrants, find \( \theta \) by calculating: - For third quadrant: \( \theta = 180^{\circ} + 34.4^{\circ} = 214.4^{\circ} \) - For fourth quadrant: \( \theta = 360^{\circ} - 34.4^{\circ} = 325.6^{\circ} \).

Step 3: Solving for \( \cos \theta = 0.7490 \)

1. Use a calculator to find the reference angle using \( \theta = \arccos(0.7490) \), which gives approximately \( \theta = 41.8^{\circ} \).2. Since cos is positive in the first and fourth quadrants, the solutions are: - First quadrant: \( \theta = 41.8^{\circ} \) - Fourth quadrant: \( \theta = 360^{\circ} - 41.8^{\circ} = 318.2^{\circ} \).

Step 4: Solving for \( \tan \theta = 2.798 \)

1. Use a calculator to find the reference angle using \( \theta = \arctan(2.798) \), which gives approximately \( \theta = 70.9^{\circ} \).2. Since tan is positive in the first and third quadrants, the solutions are: - First quadrant: \( \theta = 70.9^{\circ} \) - Third quadrant: \( \theta = 180^{\circ} + 70.9^{\circ} = 250.9^{\circ} \).

Step 5: Solving for \( \cot \theta = -0.9601 \)

1. Use a calculator to find the reference angle using \( \theta = \arccot(0.9601) \), which gives approximately \( \theta = 46.1^{\circ} \).2. Since cot is negative in the second and fourth quadrants, the solutions are: - Second quadrant: \( \theta = 180^{\circ} - 46.1^{\circ} = 133.9^{\circ} \) - Fourth quadrant: \( \theta = 360^{\circ} - 46.1^{\circ} = 313.9^{\circ} \).

Step 6: Solving for \( \sec \theta = -1.116 \)

1. Use a calculator to find the reference angle for \( \cos \theta = -1/(-1.116) = 0.8961 \), then \( \theta = \arccos(0.8961) \), which gives approximately \( \theta = 26.5^{\circ} \).2. Since sec is negative in the second and third quadrants, find \( \theta \) by calculating: - Second quadrant: \( \theta = 180^{\circ} - 26.5^{\circ} = 153.5^{\circ} \) - Third quadrant: \( \theta = 180^{\circ} + 26.5^{\circ} = 206.5^{\circ} \).

Step 7: Solving for \( \csc \theta = 1.485 \)

1. Use a calculator to find the reference angle for \( \sin \theta = 1/1.485 = 0.673 \), then \( \theta = \arcsin(0.673) \), which gives approximately \( \theta = 42.3^{\circ} \).2. Since csc (sin) is positive in the first and second quadrants, the solutions are: - First quadrant: \( \theta = 42.3^{\circ} \) - Second quadrant: \( \theta = 180^{\circ} - 42.3^{\circ} = 137.7^{\circ} \).

Key Concepts

Understanding Reference Angles in Trigonometry

When solving trigonometric equations, the reference angle plays a crucial role. A reference angle is the acute angle formed by the terminal side of an angle and the horizontal axis. It is always a positive angle and lies between 0 and 90 degrees. This concept helps us determine the solutions in different quadrants.

• To find the reference angle for an angle with a negative sine, like in step 2, you first find the positive angle using a calculator, i.e., by plugging in the positive value to \( \arcsin \).
• The reference angle is particularly useful because trigonometric functions (sine, cosine, etc.) repeat their values in cycles across the four quadrants, and these values are often symmetric or equivalent by adding or subtracting with specific quadrant values (like 180 or 360 degrees).

By utilizing reference angles, we can systematically determine the general angle solutions within a specified interval like \( \left[0^{\circ}, 360^{\circ}\right) \). This is especially helpful in analyzing wave functions and periodic patterns seen in various scientific disciplines.
Understanding and drawing reference angles can visually assist in grasping their significance in solving trigonometric equations.

Role of Trigonometric Functions in Angle Calculation

Trigonometric functions—sine, cosine, and tangent—represent relationships between the angles and sides of a triangle. They are pivotal in solving equations involving angles in trigonometry.

• The sine function, for example, is the ratio of the opposite side to the hypotenuse, while cosine represents the adjacent side to hypotenuse.
• Tangent is the ratio of the opposite to adjacent side. Other functions like cotangent, secant, and cosecant are reciprocals of tangent, cosine, and sine respectively.

In the exercise, we determine angles based on negative or positive values of these functions by using their inverse functions and deriving the reference angle. For instance, \( \arcsin \), \( \arccos \), and \( \arctan \) help retrieve an initial angle, which is adapted per quadrant rules to find potential solutions within the given range. Comprehending how these functions interact through angles increases our capability to solve equations not only in pure mathematics but also in applied fields such as physics, engineering, and computer science. Trig functions underpin periodic phenomena like sound waves and oscillations, demonstrating their broad application spectrum.

Degrees: The Measurement of Angles

Degrees are a unit of angular measurement integral to solving and understanding trigonometric equations. A full circle comprises 360 degrees, enabling us to reference any angle based on this complete rotation.

• In trigonometry, especially when dealing with problems like those in the exercise, it's vital to understand how degrees represent any angle within its range.
• Rotations can be divided into quadrants, with each quadrant encompassing 90 degrees, crucial for determining where positive and negative values of trigonometric functions occur.

By bounding solutions within \( \left[0^{\circ}, 360^{\circ}\right) \), we ensure that all calculated angles fall within one complete cycle, simplifying the problem-solving approach. Using degrees allows for intuitive understanding, as opposed to radians, which furnishes a more mathematical angle measurement. The clarity afforded by degree measures makes them particularly user-friendly right from high school trigonometry to advanced design and navigation pursuits.