Problem 23 - Precalculus Mathematics for Calculus

Question

Find all angles \(\theta\) between \(0^{\circ}\) and \(180^{\circ}\) satisfying the given equation. $$\sin \theta=0.7$$

Step-by-Step Solution

Answer

The angles are approximately \(44.4^{\circ}\) and \(135.6^{\circ}\).

Step 1: Identify the inverse function

To find the angle whose sine is 0.7, we need to use the inverse sine function. The inverse sine (or arcsine) will give us the principal value.

Step 2: Calculate the principal value

Use the inverse sine function to calculate the principal angle. We have \( \theta_1 = \sin^{-1}(0.7) \). Using a calculator, we find \( \theta_1 \approx 44.4^{\circ} \).

Step 3: Identify other possible angles

Since the sine function is positive in both the first and second quadrants, we must also consider the angle in the second quadrant. If \( \theta_1 \) is the reference angle, the second quadrant angle is \( \theta_2 = 180^{\circ} - \theta_1 \).

Step 4: Calculate the second angle

Compute the angle \( \theta_2 \) using the formula from Step 3: \( \theta_2 = 180^{\circ} - 44.4^{\circ} = 135.6^{\circ} \).

Step 5: Collect all solutions

Therefore, the angles between \(0^{\circ}\) and \(180^{\circ}\) that satisfy the equation are \(44.4^{\circ}\) and \(135.6^{\circ}\).

Key Concepts

Sine Function

The sine function is a fundamental aspect of trigonometry. It is a way to relate the angles of a right triangle to the lengths of its sides. Specifically, the sine of an angle in a right triangle is defined as the length of the side opposite the angle divided by the length of the hypotenuse. Mathematically, it is expressed as:

• For a given angle \( \theta \): \( \sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}} \)

This function is periodic, meaning it repeats its values in a regular pattern. This periodic nature is important because it allows for the sine function to have multiple angles (or solutions) for a given sine value, depending on the range we consider.
The sine function is positive in the first and second quadrants, which explains why multiple solutions are typically found in these quadrants for trigonometric problems.

Inverse Sine Function

The inverse sine function, often written as \( \sin^{-1}(x) \) or \( \arcsin(x) \), is used to find an angle when the sine of that angle is known. For instance, to find the angle \( \theta \) such that \( \sin \theta = 0.7 \), we apply the inverse sine function:

• \( \theta = \sin^{-1}(0.7) \)

The result of this operation, called the principal value, is the angle whose sine is 0.7 and typically lies between \( -90^{\circ} \) and \( 90^{\circ} \).
Since the sine function is not one-to-one (like a straight line), its inverse is restricted to specific intervals to ensure it functions properly. This principal value gives us the angle in the first quadrant.
Understanding this concept helps in solving trigonometric equations by allowing us to find the angles or solutions that correspond to a given sine value within a specific range.

Reference Angle

A reference angle is a crucial concept in trigonometry used for finding all possible solutions to trigonometric equations. It helps identify other possible angles by comparing them to a known angle in the first quadrant.
After finding an initial angle using the inverse sine function, this angle can serve as a reference to determine additional angles in other quadrants where the sine value remains the same.

• For an angle \( \theta_1 \), the reference angle is \( \theta_1 \) itself in the first quadrant.

In the previous solution, we used the reference angle \( 44.4^{\circ} \) to find another angle in the second quadrant:

• If \( \theta_1 \) is the reference angle, the second quadrant angle is computed as \( 180^{\circ} - \theta_1 \).

This approach is helpful because trigonometric functions like sine have symmetrical properties, allowing one to extend solutions to other quadrants.

Quadrants in Trigonometry

In trigonometry, the coordinate plane is divided into four quadrants. Each quadrant allows the sine, cosine, and tangent functions to take on specific positive or negative values.
The first quadrant (where both x and y are positive) and the second quadrant (where x is negative and y is positive) are critical when dealing with the sine function.

• In the first quadrant, sine is positive and increases from 0 to 1.
• In the second quadrant, sine is also positive, as it decreases from 1 back to 0 as it approaches \( 180^{\circ} \).

This symmetry in the sine function is essential when determining solutions for trigonometric equations. For instance, when required to find all angles between \( 0^{\circ} \) and \( 180^{\circ} \) that satisfy a trigonometric equation, we consider that angles in the first quadrant and complementary angles in the second quadrant can be viable solutions. Understanding these quadrants enhances our ability to solve such trigonometric problems accurately.