Problem 10 - Precalculus: Functions and Graphs

Question

Find the exact value. (a) \(\cos (5 \pi / 4) \) (b) \(\cos (-11 \pi / 6)\)

Step-by-Step Solution

Answer

(a) \(-\sqrt{2}/2\); (b) \(\sqrt{3}/2\).

Step 1: Determine the Quadrant for (a)

First, identify which quadrant the angle \(5\pi/4\) lies in. Since \(5\pi/4\) is greater than \(\pi\) (which is \(4\pi/4\)) but less than \(3\pi/2\) (which is \(6\pi/4\)), it lies in the third quadrant.

Step 2: Use Reference Angle for (a)

The reference angle for \(5\pi/4\) is \(5\pi/4 - \pi = \pi/4\). In the third quadrant, the cosine is negative.

Step 3: Calculate the Cosine for (a)

The exact value of \(\cos(\pi/4)\) is \(\sqrt{2}/2\), so the cosine of \(5\pi/4\) is \(-\sqrt{2}/2\) since it is in the third quadrant.

Step 4: Determine the Quadrant for (b)

For the angle \(-11\pi/6\), add \(2\pi\) to convert it into a positive angle: \(-11\pi/6 + 12\pi/6 = \pi/6\). \(\pi/6\) is in the first quadrant.

Step 5: Use Reference Angle for (b)

In the first quadrant, angles are already reference angles and since cosine is positive in the first quadrant, use the exact value directly.

Step 6: Calculate the Cosine for (b)

The exact value of \(\cos(\pi/6)\) is \(\sqrt{3}/2\). Hence, \(\cos(-11\pi/6) = \sqrt{3}/2\).

Key Concepts

Cosine

The cosine function is a fundamental trigonometric function that is vital in mathematics, especially in the realms of geometry and calculus. It describes the relationship between the angle of a right-angled triangle and the length of its adjacent side compared to its hypotenuse.

• The cosine of an angle \(\theta\) in a right triangle is defined as \(\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}\).
• It is also used to represent the x-coordinate of a point on the unit circle, where the point is formed by the angle between the positive x-axis and a line joining the origin to the point.
• This means, for an angle \(\theta\) measured in radians on the unit circle, the cosine function gives the horizontal distance from the origin to the circle's edge.

In the unit circle, the cosine is periodic with a period of \(2\pi\), meaning \[\cos(\theta+2\pi) = \cos(\theta)\]. This cyclic nature is crucial when solving problems involving angles larger than \(2\pi\) or negative angles. It ensures the cosine function's values repeat, which helps simplify complex angle calculations as seen in the trigonometric calculations involving positive and negative angles.

Reference Angle

A reference angle is an angle's smallest positive co-terminal angle measured from the terminal side to the x-axis. Understanding reference angles is important because they allow us to find trigonometric function values for angles outside the first quadrant using known values from the first quadrant.

• To find a reference angle for any given angle, locate how far the angle is from either \(0\), \(\pi/2\), \(\pi\), or \(3\pi/2\).
• For instance, in the given problem, the reference angle for \(5\pi/4\) is \(\pi/4\). This is calculated as \(5\pi/4 - \pi\) because \(5\pi/4\) is greater than \(\pi\) but less than \(3\pi/2\).
• Reference angles are always positive and generally fall within the range of \(0\) to \(\pi/2\) radians (or \(0º\) to \(90º\)).

Utilizing such angles simplifies the computation of trigonometric functions since reference angles correspond to standard angles whose trigonometric values are typically memorized or tabulated, like \(\pi/4, \pi/3, \pi/6\). With reference angles, you only need to adjust the sign of the function based on the angle's quadrant.

Quadrants

The coordinate plane is divided into four quadrants. These quadrants help determine the signs of trigonometric functions. Understanding which quadrant an angle lies in can help you determine the signs of different trigonometric functions, like sine, cosine, and tangent.

• The First Quadrant: All trigonometric functions are positive.
• The Second Quadrant: Sine is positive while cosine and tangent are negative.
• The Third Quadrant: Tangent is positive while sine and cosine are negative.
• The Fourth Quadrant: Cosine is positive while sine and tangent are negative.

Angles are typically measured starting from the positive x-axis, moving counter-clockwise for positive angles and clockwise for negative angles. For example, \(5\pi/4\) falls in the third quadrant, making the \cos\ value negative, while \(-11\pi/6\) adjusted to \(\pi/6\) lands in the first quadrant where the \cos\ value is positive. This understanding is crucial for correctly evaluating trigonometric functions based on their carefully assigned quadrants.