Problem 103 - A Graphical Approach to Precalculus with Limits

Question

Find the acute angle \(\theta\) that satisfies the given equation. Express your answer as an inverse trigonometric function and as the measure of \(\theta\) in degrees. $$\cos \theta=\frac{\sqrt{3}}{2}$$

Step-by-Step Solution

Answer

\( \theta = \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) = 30^\circ \)

Step 1: Identify the Trigonometric Function

We start with the equation \( \cos \theta = \frac{\sqrt{3}}{2} \). This equation expresses \( \theta \) in terms of its cosine value.

Step 2: Use the Inverse Trigonometric Function

To find \( \theta \), we apply the inverse cosine function: \( \theta = \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) \). This gives us the principal value of the angle with that specific cosine value.

Step 3: Convert to Degrees

The angle \( \theta = \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) \) can be converted to degrees. Knowing that \( \cos 30^\circ = \frac{\sqrt{3}}{2} \), we determine that \( \theta \) in degrees is \( 30^\circ \).

Step 4: Verify the Angle is Acute

An acute angle is one that is less than \( 90^\circ \). Since \( 30^\circ < 90^\circ \), we confirm that \( \theta \) is acute.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions help us find angles when given the value of a trigonometric function. For example, if we know the cosine of an angle, we can find the angle itself using the inverse cosine function, also known as the arccosine. This is often written as \( \cos^{-1} \).
The inverse cosine function, \( \cos^{-1}(x) \), outputs an angle \( \theta \) such that \( \cos(\theta) = x \).
It is important to note that the principal range of the inverse cosine function is from \( 0 \) to \( \pi \) radians, or from \( 0^\circ \) to \( 180^\circ \) degrees.
Thus, when applied to \( \frac{\sqrt{3}}{2} \), the inverse function helps identify the angle that has this specific cosine value.

• The equation \( \theta = \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) \) allows us to find \( \theta \).
• Inverse functions are very handy in trigonometry for solving equations where the angle is unknown.

Acute Angles

Acute angles are angles that measure less than \( 90^\circ \). These angles are commonly studied in geometry and trigonometry because of their special properties and frequent application in problems.
For example, when determining whether an angle is acute, we compare it to \( 90^\circ \). If the angle is less, it qualifies as acute.
In the given example, we find that the angle \( \theta \), calculated as \( 30^\circ \), is acute:

• Since \( 30^\circ < 90^\circ \), it is indeed an acute angle.
• Knowing an angle is acute can help confirm correctness when working through a problem.

Trigonometric Identities

Trigonometric identities are essential tools that allow us to verify and simplify expressions involving trigonometric functions. One commonly used identity is the Pythagorean Identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
For solving the original problem, one well-known identity was used implicitly. The fact that \( \cos 30^\circ = \frac{\sqrt{3}}{2} \) is a result of these identities.
There are several angle values where the trigonometric function values are well-known, like \( \cos 45^\circ = \frac{1}{\sqrt{2}} \) and \( \cos 60^\circ = \frac{1}{2} \).

• Trigonometric identities aid in transforming and solving equations.
• They offer a framework to understand relationships between angles and lengths in triangles.
• Knowing these identities improves the ability to solve trigonometric problems efficiently.