Problem 33 - Algebra 2 and Trigonometry

Question

In \(28-43,\) for each function value, if \(0^{\circ} \leq \theta <3 60^{\circ},\) find, to the nearest degree, two values of \(\theta\) \(\sin \theta=-0.0523\)

Step-by-Step Solution

Answer

The two values of \( \theta \) are \(183^{\circ}\) and \(357^{\circ}\).

Step 1: Understanding the Range

We're given a trigonometric equation and asked to find angles \( \theta \) within the range \( 0^{\circ} \leq \theta < 360^{\circ} \). First, we need to understand this means we're interested in angles in a full circle.

Step 2: Determine Possible Quadrants

The sine of an angle is negative, which occurs in the third and fourth quadrants of the unit circle. Therefore, we will find angles in these quadrants.

Step 3: Find the Reference Angle

To find the reference angle, calculate \( \theta = \arcsin(0.0523) \). Since \(0.0523\) is a small positive number, the reference angle \( \alpha \) is \( \arcsin(0.0523) \approx 3^{\circ} \).

Step 4: Calculate Angles in the Third and Fourth Quadrants

For the third quadrant, the angle is \( \theta = 180^{\circ} + \alpha = 180^{\circ} + 3^{\circ} = 183^{\circ} \). For the fourth quadrant, the angle is \( \theta = 360^{\circ} - \alpha = 360^{\circ} - 3^{\circ} = 357^{\circ} \).

Step 5: Confirm Answer Choices Fit Range

Both angle solutions, \(183^{\circ}\) and \(357^{\circ}\), fall within the given range \(0^{\circ} \leq \theta < 360^{\circ}\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It's simply a circle with a radius of one, centered at the origin of a coordinate plane. This makes it particularly useful for understanding angles and trigonometric functions, as the coordinates of any point on the unit circle are equal to \((\cos \theta, \sin \theta)\). In this context, the sine value \(\sin \theta = -0.0523\) means that we are looking for the angle \(\theta\) for which the y-coordinate (sine value) on the unit circle is \(-0.0523\). The full range of angles on the unit circle is from \(0^{\circ}\) to \(360^{\circ}\), effectively covering a full circle.

Reference Angle

A reference angle helps us simplify solving trigonometric equations by reducing them to acute angles. It's the smallest angle made with the horizontal axis, and it’s always positive. When you calculate the arcsin of a positive number to find a reference angle, you're finding the acute angle related to that sine value. In our exercise, since \(\arcsin(0.0523)\) is approximately \(3^{\circ}\), this is the reference angle. We use this angle to calculate the two possible \(\theta\) values in the applicable quadrants by adding or subtracting it from key angular positions like \(180^{\circ}\) or \(360^{\circ}\).

Quadrants

The coordinate plane is divided into four quadrants, each representing different ranges of the angle \(\theta\). Understanding these quadrants is crucial because each one affects the sign (positive or negative) of sine, cosine, and tangent functions:

• First Quadrant: Angles from \(0^{\circ}\) to \(90^{\circ}\), all trigonometric functions are positive.
• Second Quadrant: Angles from \(90^{\circ}\) to \(180^{\circ}\), sine is positive, cosine and tangent are negative.
• Third Quadrant: Angles from \(180^{\circ}\) to \(270^{\circ}\), sine and cosine are negative, tangent is positive.
• Fourth Quadrant: Angles from \(270^{\circ}\) to \(360^{\circ}\), sine is negative, cosine is positive, tangent is negative.

Given \(\sin \theta = -0.0523\), \(\theta\) must be in either the third or fourth quadrant where the sine values are negative. Hence, the need to compute angles \(183^{\circ}\) and \(357^{\circ}\).

Arc Functions

Arc functions, such as arcsin, arccos, and arctan, are used to find the angles when presented with a trigonometric value. They are the inverse operations of the basic trigonometric functions. For instance, \(\arcsin\) is the inverse of \(\sin\), returning the angle whose sine value is given.In our case, \(\arcsin(0.0523) \approx 3^{\circ}\). This is crucial because, while the equation has a negative sine value, calculating \(\arcsin\) gives a positive reference angle which we then use correctly within the relevant quadrants to find \(\theta\). This process demonstrates why understanding inverse trigonometric functions is essential for solving angle-finding problems. These arc functions allow us to navigate and solve trigonometric equations efficiently, especially when the angle is not easily discernible.