Problem 211 - Calculus Volume 1

Question

For the following exercises, evaluate the functions. Give the exact value. $$ \cos ^{-1}\left(\frac{\sqrt{3}}{2}\right) $$

Step-by-Step Solution

Answer

The exact value is \(\frac{\pi}{6}\).

Step 1: Understand the Function

The question asks us to evaluate the inverse cosine function: \[\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)\]The inverse cosine function, \(\cos^{-1}\), outputs the angle whose cosine is a given value.

Step 2: Find the Reference Angle

We need to find an angle \(\theta\) such that:\[\cos(\theta) = \frac{\sqrt{3}}{2}\]In trigonometry, we know that the angle whose cosine is \(\frac{\sqrt{3}}{2}\) is \(\frac{\pi}{6}\) radians.

Step 3: Verify the Range of Inverse Cosine

The range of the inverse cosine function is \([0, \pi]\). Since \(\frac{\pi}{6}\) is within this interval, it is valid for \(\cos^{-1}(x)\). This verifies that our reference angle can be the solution.

Step 4: Conclude the Evaluation

Therefore, the exact value of \(\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)\) is \(\frac{\pi}{6}\) radians.

Key Concepts

Evaluating Trigonometric Functions

Evaluating trigonometric functions, particularly the inverse ones, like \( \cos^{-1} \), revolves around determining the angle associated with a given trigonometric value. For instance, when evaluating \(\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)\), you are essentially asking: "What angle gives a cosine value of \(\frac{\sqrt{3}}{2}\)?" This process involves understanding the basic relationships between angles and their cosine values.

When solving such functions, you usually rely on known values from the unit circle or a trigonometric table. The unit circle is a helpful tool because it provides exact values for sine, cosine, and tangent functions at standard angles. Remember, inverse trigonometric functions only offer solutions within a certain range to ensure the answer is unique.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions considered true for any angle inputs. These identities allow us to simplify and evaluate functions more easily. Understanding identities like \( \cos^2(\theta) + \sin^2(\theta) = 1 \) can help solve more complex trigonometric equations by reducing them into known or simpler forms.

A powerful identity that plays a role in the evaluation of inverse trigonometric functions is related to the complementary angles, particularly: \( \cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right) \). These identities show the interrelation between trigonometric functions and can provide alternate pathways to reach the solution.

• Use identities to simplify expressions - make them easier to evaluate.
• Ensure the reference angle is within the valid range.
• Use the known values of typical angles and their trigonometric equivalents.

Radian Measure

Radian measure is an essential concept in trigonometry for expressing angles. Unlike degrees, radians provide a direct relationship between the angle and the arc length. This makes them particularly valuable in various mathematical and scientific calculations.

To convert an angle from degrees to radians, use the conversion factor \( \pi \rightarrow 180^\circ \). For instance, \(30^\circ\) in radians is \( \frac{\pi}{6} \). In evaluating inverse trigonometric functions, using radians offers an exactness that degrees may not provide, especially since the unit circle is naturally measured in radians.

• Inverse trigonometric functions often use radian measures for precise calculations.
• Understanding radian measures helps in visualizing angles on the unit circle.

Cosine Function

The cosine function is one of the fundamental trigonometric functions and is effectively the adjacent-to-hypotenuse ratio of a right triangle. In the unit circle approach, cosine corresponds to the x-coordinate of a point on the circle.

The function helps describe how the angle varies with respect to this coordinate on the unit circle. For the example \( \cos(\theta) = \frac{\sqrt{3}}{2} \), knowing this corresponds to \( \theta = \frac{\pi}{6} \) or \( \theta = 30^\circ \) helps in quickly referencing back to a standard angle well-known for its cosine.

• Relates to standard positions on the unit circle.
• Helps translate geometric problems into trigonometric terms.
• Affects motion and waves, such as sound and light, by describing periodic functions.