Problem 41 - 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 \(\theta\) \(\sin \theta=0\)

Step-by-Step Solution

Answer

The two values of \( \theta \) are \(0^{\circ}\) and \(180^{\circ}\).

Step 1: Understanding the Sine Function

The sine function, \( \sin \theta \), oscillates from \(-1\) to \(1\) as \(\theta\) varies from \(0^{\circ}\) to \(360^{\circ}\). It is \(0\) at specific angles between this range.

Step 2: Identifying Angles Where Sine is Zero

The function \( \sin \theta = 0 \) implies two main angles in the given range where this is true: at \( \theta = 0^{\circ} \) and \( \theta = 180^{\circ} \). This occurs every half rotation, or every \(180^{\circ}\).

Step 3: Confirming the Values

Verify the angles by substituting them back into the sine function and check if the conditions are satisfied: \( \sin 0^{\circ} = 0 \) and \( \sin 180^{\circ} = 0 \). Both evaluate correctly to zero.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions, commonly abbreviated as \( \sin \). It helps describe the relationship between the angles and lengths in right triangles. For any angle \( \theta \), the sine represents the ratio of the length of the opposite side to the hypotenuse in a right triangle. But its application isn't limited to triangles.

The sine function is periodic and oscillates in a smooth, wave-like pattern. Its values vary between \(-1\) and \(1\) as \(\theta\) moves through a full rotation of the circle, which is \(360^\circ\). This range shows how the sine changes from starting at zero, climbing to one at \(90^\circ\), returning to zero at \(180^\circ\), descending to \(-1\) at \(270^\circ\), and finally reaching zero again at \(360^\circ\).

Understanding the behavior of the sine wave is crucial for studying how angles influence the sine value and for solving various trigonometric problems.

Angle Measurement

Angle measurement is an important concept when dealing with trigonometric functions like the sine function. Angles can be measured in degrees or radians, but using degrees is more common, especially in introductory trigonometry.

The total degrees in a circle measure \(360^\circ\), meaning that if you start at one point on the circle's circumference and make a full loop, you will travel \(360^\circ\).

• A right angle is \(90^\circ\), which is a quarter of a circle.
• A straight angle equals \(180^\circ\), representing a half-circle rotation.
• An angle of \(360^\circ\) means a full circle.

When working with trigonometric functions, having different angle measurements can influence your calculations. However, sine values repeat every \(360^\circ\), highlighting their periodic nature.

Sine of Angles

The sine of angles is a core element in trigonometry that you'll encounter frequently. For any angle \(\theta\), the sine function, \(\sin \theta\), produces specific values depending on \(\theta\). These values are crucial for solving problems and understanding the properties of trigonometric functions.

• At \(\theta = 0^\circ\), the sine is 0.
• At \(\theta = 90^\circ\), the sine climbs to 1.
• At \(\theta = 180^\circ\), it returns to 0.
• At \(\theta = 270^\circ\), the sine dips to \(-1\).
• Back at \(\theta = 360^\circ\), the sine returns to 0.

From these values, we see that \(\sin \theta = 0\) at \(\theta = 0^\circ\), \(180^\circ\), and \(360^\circ\), confirming the function's symmetry and periodicity. These specific angles are where the sine curve intercepts the horizontal axis on a graph. Recognizing these points is crucial for identifying angles that satisfy conditions like \(\sin \theta = 0\).