Problem 62 - Precalculus : Building Concepts and Connections

Question

Find the measure of the acute angle \(\theta\) for which the sine or cosine is given. $$\sin \theta=\frac{\sqrt{3}}{2}$$

Step-by-Step Solution

Answer

\(\theta = 60^{\circ}\) or \(\theta = \frac{\pi}{3}\) radians.

Step 1: Identify known sine values

Certain angles, specifically 0, 30, 45, 60, and 90 degrees (or 0, \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{2}\) radians respectively) have known values for the sine function. Identify these known values.

Step 2: Match given sine value with its corresponding angle

\(\sin 60^{\circ} = \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\). Where both sine is positive. Therefore, \(\sin \theta = \frac{\sqrt{3}}{2}\) implies that \(\theta = 60^{\circ}\) or \(\theta = \frac{\pi}{3}\).

Step 3: Final angle value

The acute angle \(\theta\) that corresponds to \(\sin \theta = \frac{\sqrt{3}}{2}\) is 60 degrees or \(\frac{\pi}{3}\) radians.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in understanding relationships between angles and side lengths in right-angled triangles. Among the primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan), each defining a unique ratio between the sides of a right-angled triangle. The sine function specifically relates the ratio of the length of the opposite side to the hypotenuse for a given angle in a right-angled triangle.

For example, in the exercise \(\sin \theta = \frac{\sqrt{3}}{2}\), we're dealing with the sine function. This exercise taps into the student's ability to recall and match specific sine values to their corresponding angles. Familiarity with these functions is essential as they have vast applications in various fields including physics, engineering, and even geography. Enhancing our understanding of these functions helps us not only solve mathematical problems but also to comprehend the circular motion, wave phenomena, and much more.

Acute Angle Determination

The determination of an acute angle, which is any angle less than 90 degrees, is often required when dealing with trigonometric functions. Acute angles are significant since trigonometric functions for these angles are most commonly presented and used in various applications. In the given exercise, students are expected to recognize that the sine function value provided corresponds to an acute angle measurement, as values of sine for acute angles fall between 0 and 1.

By learning the specific sine values of notable acute angles - such as 30, 45, and 60 degrees - students are able to swiftly deduce that \(\sin \theta = \frac{\sqrt{3}}{2}\) must correspond to \(\theta = 60^\circ\), as it is a known sine value. Drawing the correlation between the function value and the angle is a crucial step in solving trigonometric problems and understanding how these functions define the angle within its respective quadrant.

Radian and Degree Conversion

Conversion between radians and degrees is a necessary skill in trigonometry since it allows us to navigate between the two common units of angle measurement. One complete revolution corresponds to \(360^\circ\) or \(2\pi\) radians. The conversion factor between these units is \(\pi\) radians equals \(180^\circ\), so to convert from degrees to radians, we multiply by \(\frac{\pi}{180}\), and to convert from radians to degrees, we multiply by \(\frac{180}{\pi}\).

When we determined the acute angle \(\theta\) in the exercise, understanding that the sine value \(\frac{\sqrt{3}}{2}\) corresponds to a 60-degree angle, which is the same as \(\frac{\pi}{3}\) radians, is the result of such conversion. Thus, recognizing and performing radian and degree conversion is fundamental in accurately interpreting and solving trigonometric problems across various disciplines.