Problem 22 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function at the given real number. $$ \begin{array}{llll}{\text { (a) } \sin \frac{25 \pi}{2}} & {\text { (b) } \cos \frac{25 \pi}{2}} & {\text { (c) } \cot \frac{25 \pi}{2}}\end{array} $$

Step-by-Step Solution

Answer

a) 1, b) 0, c) 0.

Step 1: Understanding the periodicity of sine and cosine

The sine and cosine functions have a periodicity of \(2\pi\), meaning \(\sin(\theta) = \sin(\theta + 2k\pi)\) and \(\cos(\theta) = \cos(\theta + 2k\pi)\) where \(k\) is an integer.

Step 2a: Simplify \(\sin \frac{25\pi}{2}\) using periodicity

First, determine how many complete cycles of \(2\pi\) are in \(\frac{25\pi}{2}\). Divide \(\frac{25\pi}{2}\) by \(2\pi\) to find the number of cycles: \[ \frac{25\pi/2}{2\pi} = \frac{25}{4}. \]The remainder is found by multiplying the fractional part by \(2\pi\): \(\frac{1}{4} \times 2\pi = \frac{\pi}{2}\).Thus, \(\sin \frac{25\pi}{2} = \sin \frac{\pi}{2} = 1\).

Step 2b: Simplify \(\cos \frac{25\pi}{2}\) using periodicity

Similarly, use the result from Step 2a. The cosine function will have the same reference angle, so\[ \cos \frac{25\pi}{2} = \cos \frac{\pi}{2} = 0. \]

Step 2c: Simplify \(\cot \frac{25\pi}{2}\)

By definition, \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\). Substitute the results from Steps 2a and 2b: \[ \cot \frac{25\pi}{2} = \frac{\cos \frac{\pi}{2}}{\sin \frac{\pi}{2}} = \frac{0}{1} = 0.\]

Step 5: Conclusion Step: Verify your calculations

Check the consistency of the periodicity for each trigonometric function. Verify that your simplifications align with the unit circle values at standard angles. This ensures precision in trigonometric calculations.

Key Concepts

Periodicity of Trigonometric Functions

Trigonometric functions like sine, cosine, and cotangent exhibit periodic behavior. This means that these functions repeat their values in regular intervals. The basic period for sine and cosine functions is \(2\pi\). It implies:

• \(\sin(\theta) = \sin(\theta + 2k\pi)\)
• \(\cos(\theta) = \cos(\theta + 2k\pi)\)

where \(k\) is an integer. This property allows us to simplify expressions like \(\sin \frac{25\pi}{2}\) and \(\cos \frac{25\pi}{2}\) by reducing them to their equivalent angles within a single period \([0, 2\pi]\). The periodicity simplifies complex angle calculations by focusing on the remainder left after dividing by the period, making calculations easier.

Sine Function

The sine function is one of the fundamental trigonometric functions. It represents the y-coordinate on the unit circle for a given angle. Sine has a range between -1 and 1 and repeats every \(2\pi\) radians.

• For example, to simplify \(\sin \frac{25\pi}{2}\), determine how many full cycles of \(2\pi\) fit into \(\frac{25\pi}{2}\). The remainder is the angle within the first cycle.
• For \(\sin \frac{25\pi}{2}\), a calculation gives a remainder of \(\frac{\pi}{2}\), hence \(\sin \frac{25\pi}{2} = \sin \frac{\pi}{2} = 1\).

Understanding how to use periodicity in conjunction with the unit circle helps in predicting sine function values at high angles.

Cosine Function

Cosine is another essential trigonometric function reflecting the x-coordinate on the unit circle for a given angle. Like sine, it has a period of \(2\pi\) and its values range between -1 and 1. To simplify angles such as \(\cos \frac{25\pi}{2}\), you again check how many \(2\pi\) fits into the angle,

• Resulting in a remainder angle of \(\frac{\pi}{2}\).
• Therefore, \(\cos \frac{25\pi}{2} = \cos \frac{\pi}{2} = 0\).

Recognizing cosine's periodic nature allows for straightforward simplification of higher order angles, using the basic cosine wave formed in \([0, 2\pi]\) radians.

Cotangent Function

The cotangent function, written as \(\cot(\theta)\), is derived as the reciprocal of the tangent function, which in turn is the ratio of \(\sin\) to \(\cos\). It can be expressed as:

• \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\).

To find \(\cot \frac{25\pi}{2}\), you apply the values obtained from the cosine and sine functions:

• \(\cot \frac{25\pi}{2} = \frac{\cos \frac{\pi}{2}}{\sin \frac{\pi}{2}} = \frac{0}{1} = 0\).

Understanding the relationship between cotangent and the other trigonometric functions allows the conversion of complex angle operations into simpler ones.

Unit Circle

The unit circle is a crucial tool in trigonometry and serves as a foundation for understanding trigonometric functions. It is a circle with a radius of 1, centered at the origin of the coordinate plane. Each point on the circle corresponds to an angle measured from the positive x-axis.

• Angles are positive in the counterclockwise direction and negative clockwise.
• The sine function gives the y-coordinate while the cosine provides the x-coordinate of a point on the unit circle corresponding to a particular angle.

For example,

• At \(\theta = \frac{\pi}{2}\), the position is (0,1) making the sine \(1\) and cosine \(0\).

This fundamental understanding helps translate complex expressions into simple unit circle positions.

Angle Simplification

Angle simplification is the process of reducing given angles to their equivalent within a single cycle or period of the unit circle, usually between \(0\) and \(2\pi\). This is immensely helpful in trigonometric problems, where large angles become cumbersome.

• For example, \(\frac{25\pi}{2}\) is reduced by finding complete periods and calculating the remainder angle.
• The expression \(\frac{25\pi}{2} \div 2\pi = \frac{25}{4}\) reveals 6 complete periods and a remainder of \(\frac{\pi}{2}\).
• Thus, the simplified angle becomes \(\frac{\pi}{2}\), which is much easier to handle.

Effectively using angle simplification aligns complex angle calculations back to fundamental trigonometric identities and cycle points like the unit circle.