Problem 37 - Precalculus: Functions and Graphs

Question

Approximate, to the nearest 0.01 radian, all angles \(\theta\) in the interval \([0,2 \pi)\) that satisfy the equation. (a) \(\sin \theta=0.4195\) (b) \(\cos \theta=-0.1207\) (c) \(\tan \theta=-3.2504\) (d) cot \(\theta=2.6815\) (e) \(\sec \theta=1.7452\) (f) \(\csc \theta=-4.8521\)

Step-by-Step Solution

Answer

(a) \(\theta \approx 0.43, 2.71\); (b) \(\theta \approx 1.69, 4.59\); (c) \(\theta \approx 1.87, 4.34\); (d) \(\theta \approx 0.36, 3.50\); (e) \(\theta \approx 0.97, 5.32\); (f) \(\theta \approx 3.35, 6.08\).

Step 1: Find Initial Angle for (a)

Use the inverse sine function to find the basic angle. \(\theta = \sin^{-1}(0.4195)\). Calculate \(\theta\) using a calculator to find \(\theta \approx 0.433\, \text{radians} \).

Step 2: Apply Symmetry for (a)

Since \(\sin(\theta)\) is positive in the first and second quadrants, \(\theta \) could also be \(\pi - 0.433 \approx 2.709\, \text{radians}.\)

Step 3: Round Results for (a)

Approximate both possible angles to two decimal places: \(\theta \approx 0.43\) and \(\theta \approx 2.71\).

Step 4: Find Initial Angle for (b)

Use the inverse cosine function to find the basic angle where cosine is negative. \(\theta = \cos^{-1}(-0.1207)\). Calculate \(\theta\) to find \(\theta \approx 1.692\, \text{radians} \).

Step 5: Apply Symmetry for (b)

Since \(\cos(\theta)\) is negative in the second and third quadrants, \(\theta \) could also be \(2\pi - 1.692 \approx 4.591\, \text{radians}.\)

Step 6: Round Results for (b)

Approximate both possible angles to two decimal places: \(\theta \approx 1.69\) and \(\theta \approx 4.59\).

Step 7: Find Initial Angle for (c)

Use the inverse tangent function to find the basic angle where tangent is negative. \(\theta = \tan^{-1}(-3.2504)\). Calculate \(\theta\) to get \(\theta \approx -1.273\, \text{radians} \).

Step 8: Adjust for Tangent Symmetry (c)

Adjust the negative angle into the first positive angle by adding \(\pi\): \(\theta = \pi - 1.273 \approx 1.869\, \text{radians} \).

Step 9: Add Period for Second solution (c)

Since the period of \(\tan(\theta)\) is \(\pi\), add \(\pi\) to find the second positive angle \(\theta = 1.869 + \pi \approx 4.343\, \text{radians} \).

Step 10: Round Results for (c)

Approximate both possible angles to two decimal places: \(\theta \approx 1.87\) and \(\theta \approx 4.34\).

Step 11: Convert Cotangent to Tangent (d)

Use the reciprocal identity \(\tan \theta = \frac{1}{\cot \theta}\), so \(\tan \theta = \frac{1}{2.6815}\) \(\approx 0.3731\).

Step 12: Find Initial Angle for (d)

Use the inverse tangent function to find \(\theta = \tan^{-1}(0.3731)\). Calculate \(\theta\) to get \(\theta \approx 0.359\, \text{radians} \).

Step 13: Consider Symmetry for Tangent (d)

Since tangent is positive in the first and third quadrants, the second angle is \(\theta + \pi = 0.359 + \pi \approx 3.500\, \text{radians} \).

Step 14: Round Results for (d)

Approximate both possible angles to two decimal places: \(\theta \approx 0.36\) and \(\theta \approx 3.50\).

Step 15: Convert Secant to Cosine (e)

Use the reciprocal identity \(\cos \theta = \frac{1}{\sec \theta}\), so \(\cos \theta = \frac{1}{1.7452}\) \(\approx 0.5731\).

Step 16: Find Initial Angle for (e)

Use the inverse cosine function to find \(\theta = \cos^{-1}(0.5731)\). Calculate \(\theta\) to get \(\theta \approx 0.965\, \text{radians} \).

Step 17: Consider Symmetry for Cosine (e)

Since cosine is positive in the first and fourth quadrants, the second angle is \(\theta = 2\pi - 0.965 \approx 5.319\, \text{radians} \).

Step 18: Round Results for (e)

Approximate both possible angles to two decimal places: \(\theta \approx 0.97\) and \(\theta \approx 5.32\).

Step 19: Convert Cosecant to Sine (f)

Use the reciprocal identity \(\sin \theta = \frac{1}{\csc \theta}\), so \(\sin \theta = \frac{1}{-4.8521}\) \(\approx -0.206\).

Step 20: Find Initial Angle for (f)

Use the inverse sine function to find \(\theta = \sin^{-1}(-0.206)\). Calculate \(\theta\) to get \(\theta \approx -0.207\, \text{radians} \).

Step 21: Adjust for Negative Angle (f)

Adjust the negative angle into the first positive angle by adding \(2\pi\): \(\theta = 2\pi + (-0.207) \approx 6.076\, \text{radians} \).

Step 22: Use Symmetry for Sin (f)

Sin is negative in the third and fourth quadrants, so the second angle is \(\pi + 0.207 \approx 3.348\, \text{radians} \).

Step 23: Round Results for (f)

Approximate both possible angles to two decimal places: \(\theta \approx 3.35\) and \(\theta \approx 6.08\).

Key Concepts

Sine Function

The sine function, denoted as \( \sin(\theta) \), is one of the fundamental trigonometric functions and is vital in the study of triangles and oscillatory phenomena. The sine of an angle \( \theta \) in a right-angled triangle is defined as the ratio of the length of the opposite side to the hypotenuse. When dealing with unit circles, the value of \( \sin(\theta) \) can be interpreted as the y-coordinate of the point on the circle corresponding to \( \theta \).

To solve equations involving the sine function, such as \( \sin(\theta) = 0.4195 \), we use the inverse sine function, \( \sin^{-1} \), to find the initial angle. In this process, understanding the symmetry of the sine function is crucial. For instance, knowing that sine is positive in both the first and second quadrants helps us identify all solutions within a specific interval, like \([0, 2\pi)\).

Cosine Function

The cosine function, \( \cos(\theta) \), is another fundamental trigonometric function. It expresses the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. For angles on the unit circle, \( \cos(\theta) \) is the x-coordinate of the corresponding point.

For equations like \( \cos(\theta) = -0.1207 \), the inverse cosine function \( \cos^{-1} \) is used to find \( \theta \). The cosine function has a property of being negative in the second and third quadrants, useful when calculating all possible angles that satisfy the equation within a given domain. Cosine's symmetry simplifies finding alternate solutions and comprehending its behavior over different quadrants.

Tangent Function

The tangent function, denoted as \( \tan(\theta) \), is the ratio of the sine function to the cosine function, or equivalently, the ratio of the opposite side to the adjacent side in a right-angled triangle. In terms of the unit circle, \( \tan(\theta) \) represents the slope of the line from the origin to a point on the circle.

Solving equations like \( \tan(\theta) = -3.2504 \) involves using the inverse tangent function \( \tan^{-1} \). Knowing the periodic nature of tangent, which repeats every \( \pi \), aids in finding multiple solutions of such equations. The tangent function is negative in the second and fourth quadrants, guiding us to all relevant angle solutions within any specified interval.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions include secant, cosecant, and cotangent, which are inverse relatively to the cosine, sine, and tangent functions. Understanding these functions is crucial as they frequently appear in complex equations.

For instance, \( \sec(\theta) = 1.7452 \) translates to the cosine function as \( \cos(\theta) = \frac{1}{1.7452} \), while \( \csc(\theta) = -4.8521 \) translates to the sine function as \( \sin(\theta) = \frac{1}{-4.8521} \).

• Secant, \( \sec(\theta) \): Reciprocal of cosine, \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
• Cosecant, \( \csc(\theta) \): Reciprocal of sine, \( \csc(\theta) = \frac{1}{\sin(\theta)} \).
• Cotangent, \( \cot(\theta) \): Reciprocal of tangent, \( \cot(\theta) = \frac{1}{\tan(\theta)} \).

Utilizing reciprocal relationships simplifies the solving process and reveals how trigonometric functions are interconnected.

Angle Approximation

In trigonometry, angle approximation refers to the method of finding approximate values of angles that satisfy certain conditions, given that exact solutions can be complex or impossible to determine without technology.

For problems such as finding angles for \( \sin(\theta) = 0.4195 \), angle approximation helps in providing solutions in a manageable form like rounded to two decimal places. This technique is crucial when angles have to be expressed in radians within the range \([0, 2\pi)\). Using a calculator with inverse trigonometric functions allows for efficient finding of the basic angles.

Once the initial approximations are found, applying the symmetry and periodic properties of trigonometric functions assists in calculating all potential solutions of the given trigonometric equation within its specified interval.