Problem 65 - Calculus for Biology and Medicine

Question

Evaluate the following expressions without using a calculator: (a) \(\sin \left(-\frac{5 \pi}{4}\right)\) (b) \(\cos \left(\frac{5 \pi}{6}\right)\) (c) \(\tan \left(\frac{\pi}{3}\right)\)

Step-by-Step Solution

Answer

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

Step 1: Identify Sine of Negative Angle

Recall that the sine function is odd, which means that \( \sin(-x) = -\sin(x) \). Thus, \( \sin \left(-\frac{5\pi}{4}\right) = -\sin \left(\frac{5\pi}{4}\right) \).

Step 2: Locate Angle on the Unit Circle for Sine

Convert \( \frac{5\pi}{4} \) into degrees to understand its position on the unit circle. This is equivalent to 225°. The reference angle for 225° is 45°, which is in the third quadrant where sine is negative, meaning \( \sin \left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \).

Step 3: Evaluate Sine Function

Using the property of the sine function being odd, \( \sin \left(-\frac{5\pi}{4}\right) = -(-\frac{\sqrt{2}}{2}) = \frac{\sqrt{2}}{2} \).

Step 4: Determine Cosine for Given Angle

Convert \( \frac{5\pi}{6} \) into degrees, which is 150°. This falls in the second quadrant, where cosine is negative. The reference angle is 30° where \( \cos(30°) = \frac{\sqrt{3}}{2} \). Thus, \( \cos(150°) = -\frac{\sqrt{3}}{2} \).

Step 5: Evaluate Cosine Expression

We then find \( \cos \left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2} \).

Step 6: Identify Tangent Function

For \( \tan \left(\frac{\pi}{3}\right) \), convert \( \frac{\pi}{3} \) into degrees, which equals 60°. The tangent of 60° is \( \sqrt{3} \).

Step 7: Evaluate Tangent Expression

So, \( \tan \left(\frac{\pi}{3}\right) = \sqrt{3} \).

Key Concepts

Unit Circle

The Unit Circle is an essential concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. This circle is fundamental because it provides a way to obtain the sine, cosine, and tangent values for different angles:

• The circle's circumference allows the angle to be measured in radians, which is more common in trigonometry than degrees.
• Each point on the unit circle corresponds to \(x\) and \(y\) coordinates representing \( ext{cos}( heta)\) and \( ext{sin}( heta)\) respectively, for an angle \(\theta\).
• The tangent is given by the ratio of sine to cosine, \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).

The unit circle helps visualize where each trigonometric function is positive or negative across its four quadrants, providing a handy reference for solving trigonometric equations.

Sine Function

The Sine Function, denoted as \(\sin(x)\), is the vertical coordinate of a point on the unit circle.

• This function is periodic with a period of \(2\pi\), meaning it repeats every \(2\pi\) radians.
• It is an odd function which means \(\sin(-x) = -\sin(x)\).
• The sine values vary between \(-1\) and \(1\), peaking at \(\pi/2\) and reaching a minimum at \(3\pi/2\).

For the original exercise, finding \(\sin\left(-\frac{5\pi}{4}\right)\), involves recognizing that \(\frac{5\pi}{4}\) lies in the third quadrant where sine is negative. Since sine is an odd function, \(\sin\left(-\frac{5\pi}{4}\right) = -(-\frac{\sqrt{2}}{2}) = \frac{\sqrt{2}}{2}\).

Cosine Function

The Cosine Function, represented as \(\cos(x)\), describes the horizontal coordinate on the unit circle:

• Similar to sine, the cosine function is periodic with a \(2\pi\) period.
• Unlike sine, cosine is an even function, following \(\cos(-x) = \cos(x)\).
• Values range from \(-1\) to \(1\), with \(1\) at \(0\) and \(2\pi\) radians, and \(-1\) at \(\pi\).

In the exercise, to find \(\cos\left(\frac{5\pi}{6}\right)\), you need to recognize that \(\frac{5\pi}{6}\) rests in the second quadrant, making cosine negative. With a reference angle of 30°, and since \(\cos(30°) = \frac{\sqrt{3}}{2}\), thus \(\cos(150°) = -\frac{\sqrt{3}}{2}\).

Tangent Function

The Tangent Function, \(\tan(x)\), is the ratio of sine to cosine. This function is quite useful:

• It's periodic, with a smaller cycle of \(\pi\) due to its repeating nature.
• The function is undefined when cosine is zero, such as at \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\).
• Tangent values can be any real number, unlike sine and cosine.

In the given problem, to evaluate \(\tan\left(\frac{\pi}{3}\right)\), knowing that \(\frac{\pi}{3}\) is 60° helps. With \(\tan(60°) = \sqrt{3}\), you get \(\tan\left(\frac{\pi}{3}\right) = \sqrt{3}\).

Reference Angles

Reference angles are a key trigonometric concept helping simplify complex calculations. They allow us to refer back to key known angles in the first quadrant:

• A reference angle is the acute angle formed by the terminal side of a given angle and the horizontal axis.
• They are helpful because they exhibit the same trigonometric values (ignoring signs) as angles from other quadrants.
• Signs differ based on which quadrant the original angle is located.

For example, if analyzing the angle like \(\frac{5\pi}{4}\), in the third quadrant, its reference angle is 45°. Since the sine in the third quadrant is negative, this transforms \(\sin\left(\frac{5\pi}{4}\right)\) into \(-\frac{\sqrt{2}}{2}\). Reference angles help naturally convert angles to known trigonometric values, thus simplifying complex calculations.