Problem 102 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) $$\cos (-15 \pi / 14)$$

Step-by-Step Solution

Answer

After following steps 1-3 in order, the final answer after computing cos(\(\pi/14\)) rounded to four decimal places using a calculator will be the required solution.

Step 1: Understanding Negative Angles

The cosine of a negative angle (-θ) is the same as the cosine of the positive angle (θ). This is because cosine is an even function, which means \(\cos(-\theta) = \cos(\theta)\). So, \(\cos(-15π/14)\) is the same as \(\cos(15π/14)\).

Step 2: Accounting for the Periodicity of Cosine

The cosine function is periodic with a period of \(2\pi\). So, we find an equivalent angle within the range of [0, \(2\pi\)). Start by adding \(2\pi\) to the angle until it falls within this range. For \(15\pi/14\), we can subtract it by \(2\pi\) or \(14\pi/7\) once to obtain the angle within the desired range, so we get, \(15\pi/14 - 14\pi/7 = \pi/14\)

Step 3: Evaluating the Cosine Function Using a Calculator

When using a calculator, ensure it's set in radian mode. Enter the equivalent angle measured in radians (\(\pi/14\)) to get the value of cosine.

Key Concepts

Cosine Function

The cosine function is one of the primary functions in trigonometry, alongside sine and tangent. It is widely used to calculate the relationship between the sides and angles of a right triangle. The function is defined based on the unit circle, where the cosine of an angle corresponds to the x-coordinate of the point where the terminal side of the angle intersects the unit circle. In simpler terms,

• Cosine measures the horizontal distance from the origin to a point on the unit circle.
• It helps in figuring out the adjacent side length of a right triangle given the hypotenuse, using the formula \( ext{cos} \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\).

Understanding this function is crucial because it is mathematically expressed as an even function. This means that for any angle \(\theta\), the cosine of \(-\theta\) is equal to the cosine of \(\theta\). Knowing this, we can simplify calculations for negative angles efficiently.

Negative Angles

In trigonometry, negative angles are handled by considering their direction of measurement. While positive angles are counter-clockwise, negative angles are measured clockwise from the positive x-axis. However, the magic lies in the behavior of trigonometric functions like cosine.

• Cosine, being an even function, treats negative angles in the same way it does positive angles.
• This symmetry implies that \(\cos(-\theta) = \cos(\theta)\).

For example, when you encounter \(\cos(-15\pi/14)\), it simplifies to \(\cos(15\pi/14)\). This property allows you to disregard the negative sign, making evaluations simpler and shifting focus to the period and value of the angle's cosine.

Periodicity of Trigonometric Functions

Trigonometric functions are known for their periodic nature, making them predictable and easy to work with over repetitive cycles. Specifically, the cosine function has a period of \(2\pi\), meaning after every \(2\pi\), the function's values repeat themselves. Here's why that’s useful:

• Periodic functions allow angles to be reduced to equivalent angles within \([0, 2\pi)\).
• This simplification doesn't change the function's value, making calculations easier and more straightforward.

For the calculation \(\cos(15\pi/14)\), involving an angle exceeding \(2\pi\), periodicity helps shrink it into a simpler angle \(\pi/14\), while maintaining the same cosine value. Utilizing a calculator, always ensure it is set to the correct mode: in this case, radians, before calculating \(\cos(\pi/14)\) to get the accurate result.