Problem 18 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function at the given real number. $$ \begin{array}{llll}{\text { (a) } \sin \frac{5 \pi}{4}} & {\text { (b) } \sec \frac{5 \pi}{4}} & {\text { (c) } \tan \frac{5 \pi}{4}}\end{array} $$

Step-by-Step Solution

Answer

(a) \(-\frac{\sqrt{2}}{2}\), (b) \(-\sqrt{2}\), (c) 1.

Step 1: Identify the angle location

First, let's find out where the angle \( \frac{5\pi}{4} \) is located on the unit circle. \( \frac{5\pi}{4} \) equals \( 225^\circ \), which places it in the third quadrant.

Step 2: Determine reference angle

The reference angle for \( \frac{5\pi}{4} \) is found by subtracting \( \pi \) (or \( 180^\circ \)) from \( \frac{5\pi}{4} \), giving a reference angle of \( \frac{\pi}{4} \) (or \( 45^\circ \)).

Step 3: Calculate \( \sin \frac{5\pi}{4} \)

In the third quadrant, the sine function is negative. Therefore, \( \sin \left( \frac{5\pi}{4} \right) = -\frac{\sqrt{2}}{2} \).

Step 4: Calculate \( \sec \frac{5\pi}{4} \)

The secant function is the reciprocal of cosine. In the third quadrant, cosine is also negative. Thus, \( \cos \left( \frac{5\pi}{4} \right) = -\frac{\sqrt{2}}{2} \), so \( \sec \left( \frac{5\pi}{4} \right) = -\sqrt{2} \).

Step 5: Calculate \( \tan \frac{5\pi}{4} \)

The tangent function is positive in the third quadrant since it is the ratio of sine to cosine. \( \tan \left( \frac{5\pi}{4} \right) = \frac{\sin \left( \frac{5\pi}{4} \right)}{\cos \left( \frac{5\pi}{4} \right)} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1 \).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry and is used to understand the trigonometric functions based on angles. It is a circle with a radius of 1 and is centered at the origin of a coordinate plane.

• The unit circle helps to define the values of the sine, cosine, and tangent functions for different angles.
• Angles on the unit circle are measured in radians, where the entire circle is \(2\pi\) radians (or \(360^\circ\)).
• The position of an angle around the circle indicates which quadrant it falls in, crucial for determining the sign of trigonometric functions.

For \(\frac{5\pi}{4}\), this angle is located in the third quadrant on the unit circle. Understanding this position helps us determine the signs of the sine, cosine, and tangent functions at that angle.

Reference Angle

A reference angle is the acute angle formed by the terminal side of a given angle and the horizontal axis. It provides a way to simplify the understanding of angles greater than \(90^\circ\) or \(\pi/2\) radians.

• The reference angle is always positive and less than or equal to \(90^\circ\) (or \(\pi/2\) radians).
• It helps in determining the trigonometric function values of angles in any quadrant by considering only the acute angle.
• For any angle \(\theta\), its reference angle in the unit circle is determined by the quadrant it lies in. For instance, in the third quadrant, you would subtract \(\pi\) from the angle.

In the case of \(\frac{5\pi}{4}\), it is located in the third quadrant and its reference angle is \(\frac{\pi}{4}\), which is equivalent to \(45^\circ\). This helps in computing its exact trigonometric values.

Sine Function

The sine function relates to the vertical component or the y-coordinate on the unit circle. It represents how "high" you are above the x-axis at any given angle.

• The sine function ranges from -1 to 1. This is because the unit circle has a maximum radius of 1.
• In different quadrants of the unit circle, the sign of the sine value changes. It is positive in the first and second quadrants, and negative in the third and fourth quadrants.
• To find \(\sin\left(\frac{5\pi}{4}\right)\), recognize that it is in the third quadrant where sine values are negative.

Therefore, \(\sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}\). The reference angle \(\frac{\pi}{4}\) or \(45^\circ\) tells us the absolute value, which is \(\frac{\sqrt{2}}{2}\).

Cosine Function

Cosine is another primary trigonometric function that relates the horizontal component or the x-coordinate on the unit circle to an angle.

• Just like sine, the cosine function also ranges from -1 to 1.
• The cosine function is positive in the first and fourth quadrants and negative in the second and third quadrants.
• For an angle like \(\frac{5\pi}{4}\), found in the third quadrant, the cosine will be negative.

Thus, \(\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}\). This closely mirrors the sine function for angles with a reference angle of \(\frac{\pi}{4}\). In this situation, the absolute value of cosine is also \(\frac{\sqrt{2}}{2}\).

Tangent Function

Tangent is a significant trigonometric function derived from sine and cosine. It is defined as the ratio of sine to cosine.

• The tangent function can take on any value from \(-\infty\) to \(\infty\).
• The sign of the tangent is determined by the quadrant of the angle. It is positive in the first and third quadrants, and negative in the second and fourth quadrants.
• In the case of \(\frac{5\pi}{4}\), the angle lies in the third quadrant, making the tangent function positive.

Therefore, \(\tan\left(\frac{5\pi}{4}\right)\) is the ratio of \(\sin\left(\frac{5\pi}{4}\right)\) to \(\cos\left(\frac{5\pi}{4}\right)\), which results in 1. This value arises because both sine and cosine for \(\frac{\pi}{4}\) have the same absolute value.