Problem 99 - Algebra and Trigonometry

Question

find two values of \(\theta, 0 \leq \theta<2 \pi,\) that satisfy each equation. $$ \sin \theta=\frac{\sqrt{2}}{2} $$

Step-by-Step Solution

Answer

The two values of \(\theta\) that satisfy the equation are \(\theta = \pi/4\) and \(\theta = 3\pi/4\).

Step 1: Understand the quadrant range

Recall the unit circle in trigonometry. The sine of an angle is positive in both the first quadrant (0 to \(\pi/2\)) and the second quadrant (\(\pi/2\) to \(\pi\)). Thus, there will be one solution for \(\theta\) in each of these quadrants.

Step 2: Determine the corresponding angles

The sin values are equal to the y-coordinates of the points on the unit circle. In this case, the values of \(\theta\) that give a sine of \(\frac{\sqrt{2}}{2}\) are \(\pi/4\) (45 degrees) and \(3\pi/4\) (135 degrees) respectively. These are the two angle values in the range \(0\leq \theta < 2 \pi\) that make \(\sin \theta=\frac{\sqrt{2}}{2}\).

Step 3: Check your answers

Verify your answers by substituting \(\theta = \pi/4\) and \(3\pi/4\) back into the original equation to confirm that \(\sin \theta = \frac{\sqrt{2}}{2}\).

Key Concepts

Unit Circle

The unit circle is a crucial concept in trigonometry. It is a circle with a radius of one, centered at the origin of a coordinate plane. This circle allows us to define trigonometric functions geometrically. Every point \((x, y)\) on the unit circle corresponds to an angle \(\theta\), measured from the positive x-axis. This angle is usually measured in radians, a unit of angular measure.

• The key feature of the unit circle is that for any angle \(\theta\), the coordinates \((x, y)\) on the circle give us the cosine and sine of that angle: \(\cos \theta = x\) and \(\sin \theta = y\).
• Thus, the y-coordinate in this scenario represents the sine value of angle \(\theta\).

In quadrants, sine is positive in the first and second quadrants of the unit circle. This is relevant for solving trigonometric equations because it helps determine where the angle's sine is positive, as seen in our exercise.

Sine Function

The sine function is a fundamental trigonometric function that relates to angles in a right triangle. In the context of the unit circle, it relates to the y-coordinates of the circle's points.

• The sine of an angle \(\theta\) is defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle. On the unit circle, the hypotenuse is always 1, making the sine the y-coordinate directly.
• It is periodic and repeats its values in regular intervals of \(2\pi\) radians.

In the exercise provided, we specifically look for angles that produce a sine value of \(\frac{\sqrt{2}}{2}\). For the unit circle, these angles are known and correspond to the special angles \(\frac{\pi}{4}\) and \(\frac{3\pi}{4}\). These particular angles are commonly used due to their symmetrical positions along the circle.

Radians

Radians are an alternative to degrees for measuring angles. This measurement is based on the radius of a circle.

• One complete revolution around a circle is \(2\pi\) radians. Therefore, \(\pi\) radians is equivalent to 180 degrees.
• This makes \(\pi/4\) radians equal to 45 degrees, and \(3\pi/4\) radians equal to 135 degrees.
• Using radians simplifies many mathematical equations, especially in calculus, as it aligns the measurement of angles with the arc length on circles.

Understanding radians is critical when working with trigonometric equations, such as in the solution of \(\sin \theta = \frac{\sqrt{2}}{2}\). The angles \(\theta\) are presented in radians to seamlessly connect with the concept of the unit circle and sinusoidal functions.