Problem 381 - Precalculus

Question

Find the missing angle: \(\cos \left(\frac{\pi}{6}\right)=\sin (\\_)\)

Step-by-Step Solution

Answer

The missing angle is \(\frac{\pi}{3}\).

Step 1: Understand the problem

We need to find the angle whose sine is equal to the cosine of \(\frac{\pi}{6}\). We know that \(\cos(\theta) = \sin(\frac{\pi}{2} - \theta)\), where \(\theta\) is the initial angle given.

Step 2: Calculate \(\cos\left(\frac{\pi}{6}\right)\)

The value of \(\cos\left(\frac{\pi}{6}\right)\) is a standard trigonometric value. It is equal to \(\frac{\sqrt{3}}{2}\).

Step 3: Determine the equivalent sine angle

We need to find an angle \(x\) such that \(\sin(x) = \frac{\sqrt{3}}{2}\). According to the identity \(\cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right)\), it follows that \(x = \frac{\pi}{2} - \frac{\pi}{6}\).

Step 4: Calculate the missing angle

Simplify \(\frac{\pi}{2} - \frac{\pi}{6}\) to find \(x\).- Rewrite \(\frac{\pi}{2}\) as \(\frac{3\pi}{6}\) so the subtraction is easier: \[ \frac{3\pi}{6} - \frac{\pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3} \] Thus, the missing angle \(x\) is \(\frac{\pi}{3}\).

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It describes the adjacent side length over the hypotenuse in a right-angled triangle. For any angle \( \theta \) on the unit circle, the cosine value corresponds to the x-coordinate of the point where the terminal side of the angle intersects the circle. This makes cosine particularly useful for calculations involving horizontal distances or projections.

Here are some key points:

• The range of the cosine function is from -1 to 1. This means it can never produce a value outside this interval.
• Cosine is an even function, meaning \( \cos(\theta) = \cos(-\theta) \). This symmetry simplifies calculations for angles on opposite sides of the origin.
• It has a period of \(2\pi\), meaning the function repeats every \(2\pi\) radians.
• The cosine of standard angles can be memorized or calculated easily, e.g., \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \).

Sine Function

The sine function is another basic trigonometric function, representing the ratio of the opposite side to the hypotenuse in a right-angled triangle. It's instrumental for calculating vertical distances or projections. On the unit circle, sine corresponds to the y-coordinate for a given angle \(\theta\).

Notable characteristics include:

• The sine function also ranges from -1 to 1, with outputs corresponding to possible y-values on the unit circle.
• Sine is an odd function, implying that \( \sin(-\theta) = -\sin(\theta) \), which helps in simplifying expressions involving negative angles.It is periodic, repeating every \(2\pi\) radians just like cosine.
• Standard angle values, such as \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \), help in quickly solving trigonometry problems.

Angle Subtraction Identity

Trigonometric identities are tools that provide different ways to express trigonometric functions. The angle subtraction identity is vital among these. It's particularly used to express cosine and sine in terms of angles summed or subtracted from each other.

The identity \( \cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta) \) helps evaluate the difference of two angles:

• For small angles or non-standard angles, this identity aids in simplifying expressions to find exact values.
• In the solution provided, we used a derivative of this identity. Specifically, \( \cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right) \), to find equivalent sine values for a given cosine.
• This is crucial because it connects the core trigonometric functions by expressing them with shifted angles, proving especially helpful with the unit circle relationships.

Understanding these identities enhances comprehension of how angles and their trigonometric outcomes interrelate.