Problem 36 - Precalculus: Functions and Graphs

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) \(\sin \theta=0.8225\) (b) \(\cos \theta=-0.6604\) (c) \(\tan \theta=-1.5214\) (d) cot \(\theta=1.3752\) (e) \(\sec \theta=1.4291\) (f) csc \(\theta=-2.3179\)

Step-by-Step Solution

Answer

(a) 55.4°, 124.6°; (b) 131.4°, 228.6°; (c) 123.8°, 303.8°; (d) 36.2°, 216.2°; (e) 45.7°, 314.3°; (f) 205.6°, 334.4°.

Step 1: Solve part (a) using arcsin

Determine the angle whose sine is 0.8225 using the inverse sine function. Calculate \( \theta_1 = \arcsin(0.8225) \approx 55.4^{\circ} \).Since sine is positive in both the first and second quadrants, another angle that satisfies the equation is:\[ \theta_2 = 180^{\circ} - \theta_1 \approx 124.6^{\circ} \].

Step 2: Solve part (b) using arccos

Determine the angle whose cosine is -0.6604 using the inverse cosine function. Calculate \( \theta_1 = \arccos(-0.6604) \approx 131.4^{\circ} \).Since cosine is negative in the second and third quadrants, another angle that satisfies the equation is:\[ \theta_2 = 360^{\circ} - \theta_1 \approx 228.6^{\circ} \].

Step 3: Solve part (c) using arctan

Determine the angle whose tangent is -1.5214 using the inverse tangent function. Calculate \( \theta_1 = \arctan(-1.5214) \approx -56.2^{\circ} \).Adjust \( \theta_1 \) to fall within the interval by adding 180° to find the correct angle in the fourth quadrant:\[ \theta_1 = 180^{\circ} - 56.2^{\circ} \approx 123.8^{\circ} \].Another angle, in the fourth quadrant, is:\[ \theta_2 = 360^{\circ} - 56.2^{\circ} \approx 303.8^{\circ} \].

Step 4: Solve part (d) using arccot

Determine the angle whose cotangent is 1.3752 using the inverse cotangent function.Convert cotangent to tangent: \( \tan \theta = \frac{1}{1.3752} \approx 0.7275 \).Calculate \( \theta_1 = \arctan(0.7275) \approx 36.2^{\circ} \).Since cotangent is positive in the first and third quadrants, the other angle is:\[ \theta_2 = 180^{\circ} + \theta_1 \approx 216.2^{\circ} \].

Step 5: Solve part (e) using arcsec

Determine the angle whose secant is 1.4291 using the inverse secant function.Convert secant to cosine: \( \cos \theta = \frac{1}{1.4291} \approx 0.6998 \).Calculate \( \theta_1 = \arccos(0.6998) \approx 45.7^{\circ} \).Since secant is positive in the first and fourth quadrants, the other angle is:\[ \theta_2 = 360^{\circ} - \theta_1 \approx 314.3^{\circ} \].

Step 6: Solve part (f) using arccsc

Determine the angle whose cosecant is -2.3179 using the inverse cosecant function.Convert cosecant to sine: \( \sin \theta = \frac{-1}{2.3179} \approx -0.4315 \).Calculate \( \theta_1 = \arcsin(-0.4315) \approx -25.6^{\circ} \).Adjust \( \theta_1 \) to fall within the interval by adding 360° to make it positive:\[ \theta_1 = 334.4^{\circ} \].Since sine is negative in the third and fourth quadrants, the other angle is:\[ \theta_2 = 180^{\circ} - (-25.6^{\circ}) \approx 205.6^{\circ} \].

Key Concepts

Angle Approximation

When dealing with trigonometric functions, we often encounter the need to approximate angles using inverse trigonometric functions. These functions allow us to find an angle when given a specific value of a trigonometric ratio. For example, when provided with a value for \(\sin \theta = 0.8225\), we use the arcsin function to find the angle \( \theta \). This involves calculating \( \theta = \arcsin(0.8225) \) which gives us an approximate angle of \(55.4^{\circ}\).
It is important to remember that the basic trigonometric functions repeat their values in specific quadrants, making it crucial to determine all possible angle solutions within a given range. Here, the sin function is positive in the first and second quadrants. Therefore, another solution is \( \theta = 180^{\circ} - 55.4^{\circ} \approx 124.6^{\circ}\).

• Use the inverse function corresponding to the known trigonometric ratio.
• Remember that angles can have multiple solutions depending on their quadrant.
• Approximate angles to the nearest desired unit for practicality.

This systematic approach helps in accurately approximating angles in problems dealing with trigonometric functions.

Trigonometric Equations

Solving trigonometric equations involves using known values to find unknown angles. These equations typically involve the use of inverse trigonometric functions to isolate the variable denoting the angle. Take for example the equation \( \cos \theta = -0.6604 \). By using the arccos function, you determine \( \theta = \arccos(-0.6604) \approx 131.4^{\circ} \).
It’s necessary to consider the characteristics of trigonometric functions that repeat their value across different quadrants:

• \( \cos \) is negative in the second and third quadrants, indicating two possible angles that solve the equation: \( \theta = 131.4^{\circ} \) and \( \theta = 360^{\circ} - 131.4^{\circ} \approx 228.6^{\circ} \).
• For functions like \( \tan \) and \( \cot \), these equations have solutions in both positive and negative ranges requiring conversions or adjustments.

Through proper understanding of how these equations function and the symmetry within their solutions, solving trigonometric equations becomes approachable.

Trigonometric Identities

Trigonometric identities are mathematical equations that involve trigonometric functions holding true for all values of the variables they include. They serve as tools for simplifying and solving complex trigonometric equations. Common identities like \( \sin^2 \theta + \cos^2 \theta = 1 \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) are often used in manipulating complex expressions.
Let's consider solving for \( \theta \) given \( \sec \theta = 1.4291 \). First, convert from secant to cosine using the identity \( \cos \theta = \frac{1}{\sec \theta} \), which results in \( \cos \theta = 0.6998 \).
Using these identities:

• Allows us to switch between trigonometric functions easily.
• Simplifies the calculation of angles from complex trigonometric functions through already known values.
• Provides broader insight into problem-solving strategies within trigonometric contexts.

Incorporating these identities effectively enables the transformation of more intricate trigonometric problems into simpler, solvable forms.