Problem 20 - Calculus Single Variable

Question

\(\theta\) is a number between 0 and \(\pi / 2\). Calculate the unevaluated trigonometric function from the given information. \(\cos (\theta+\pi) ; \cos (\theta)=0.1\)

Step-by-Step Solution

Answer

\( \cos(\theta + \pi) = -0.1 \).

Step 1: Identify Known Values

We are given \( \cos(\theta) = 0.1 \) and need to find \( \cos(\theta + \pi) \). We also know \( \pi \) radians is a half-turn.

Step 2: Apply the Trigonometric Identity

Use the identity \( \cos(\theta + \pi) = -\cos(\theta) \). This identity transforms the cosine of an angle added to \( \pi \) radians to the negative cosine of the original angle, because adding \( \pi \) shifts the angle to the opposite direction on the unit circle.

Step 3: Substitute Known Values into the Identity

Substitute the given value \( \cos(\theta) = 0.1 \) into the identity: \( \cos(\theta + \pi) = -0.1 \).

Step 4: Simplify the Expression

The calculation yields the value of \( \cos(\theta + \pi) = -0.1 \), which is our final result.

Key Concepts

Angle Transformation

Angle transformation is a valuable concept in trigonometry that involves changing an angle by adding or subtracting a constant, like \( \pi \). By recognizing how angles transform, we can easily modify known trigonometric values to uncover new ones.
When you add \( \pi \) to an angle, such as in \( \theta + \pi \), you're effectively rotating the angle halfway around the unit circle. This transformation has a key effect on trigonometric functions:

• The cosine function becomes its negative counterpart, which means \( \cos(\theta + \pi) = -\cos(\theta) \). This reflects how the unit circle positions have changed on the horizontal axis as the angle shifts along the circle.

By using these simple yet powerful transformations, you can solve complex trigonometry problems with less computation.

Unit Circle

The unit circle is a crucial tool in trigonometry, representing all angle measures from 0 to \( 2\pi \) radians or 0 to 360 degrees on a circle with a radius of one. Every point on the unit circle corresponds to an angle and is associated with a cosine value (horizontal component) and a sine value (vertical component).
Knowing the location of angles on the unit circle helps you quickly ascertain the signs and values of trigonometric functions:

• A 0 to \( \pi/2 \) angle lies in the first quadrant where both cosine and sine values are positive.
• After adding \( \pi \) to an angle, the resulting position is diagonally across the circle from the original position, aligning it with the third quadrant where cosine and sine values are negative.

Understanding these quadrant shifts allows calculation of transformed trigonometric values effortlessly.

Cosine Function

The cosine function is one of the basic trigonometric functions, expressing the horizontal distance from the origin to a point on the unit circle defined by an angle. For an angle \( \theta \) in the first quadrant, where \( 0 \leq \theta < \pi/2 \), \( \cos(\theta) > 0 \).
For example, if \( \cos(\theta) = 0.1 \), it implies the angle is slightly near the y-axis, making the cosine value relatively small. Given the identity \( \cos(\theta + \pi) = -\cos(\theta) \), we can predict the cosine value after an angle transformation.
This identity reflects how adding \( \pi \) shifts the point across to the negative horizontal component, effectively mirroring it over the unit circle's y-axis. As a result, the cosine value changes sign, becoming negative. These insights empower you to tackle trigonometric problems with confidence by leveraging properties of the cosine function.