Problem 69 - A Graphical Approach to Precalculus with Limits

Question

Find exact values of the six trigonometric functions for each angle by hand. Do not use a calculator. $$-\frac{7 \pi}{4}$$

Step-by-Step Solution

Answer

All trigonometric functions for \(-\frac{7\pi}{4}\) are positive: sine, cosine = \(\frac{\sqrt{2}}{2}\); tangent, cotangent = 1; secant, cosecant = \(\sqrt{2}\).

Step 1: Find the Reference Angle

The reference angle for the angle given in radians is found by first adding or subtracting multiples of \(2\pi\) until the resulting angle is between \(0\) and \(2\pi\). Here, \(-\frac{7\pi}{4} + 2\pi = \frac{\pi}{4}\). Therefore, the reference angle is \(\frac{\pi}{4}\).

Step 2: Identify the Quadrant

The angle \(-\frac{7\pi}{4}\) is equivalent to \(\frac{\pi}{4}\) in the positive direction after adding \(2\pi\), which is in the first quadrant where all trigonometric functions are positive.

Step 3: Calculate Sine and Cosine Functions

The sine and cosine of \(\frac{\pi}{4}\) are both \(\frac{\sqrt{2}}{2}\). Hence, \(\sin(-\frac{7\pi}{4}) = \frac{\sqrt{2}}{2}\) and \(\cos(-\frac{7\pi}{4}) = \frac{\sqrt{2}}{2}\).

Step 4: Calculate Tangent and Cotangent Functions

The tangent is the sine divided by the cosine. Hence, \(\tan(-\frac{7\pi}{4}) = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1\). The cotangent is the reciprocal of tangent, so \(\cot(-\frac{7\pi}{4}) = 1\).

Step 5: Calculate Secant and Cosecant Functions

The secant is the reciprocal of cosine: \(\sec(-\frac{7\pi}{4}) = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}\). The cosecant is the reciprocal of sine: \(\csc(-\frac{7\pi}{4}) = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}\).

Key Concepts

Angles in Radians

Radians are a way to measure angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 equal parts, radians use the circle's radius to set the measure. One complete revolution around a circle is equal to \(2\pi\) radians, equivalent to 360 degrees. Therefore, \( \pi \) radians correspond to 180 degrees.

• To convert from degrees to radians, multiply the degree value by \(\frac{\pi}{180}\).
• To convert from radians to degrees, multiply the radian measure by \(\frac{180}{\pi}\).

Understanding radians is crucial because most trigonometric formulas and identities assume angles are in radian measure. This unit is commonly used in higher mathematics because it simplifies various mathematical operations.

Reference Angle

The reference angle is the smallest angle that the terminal side of a given angle makes with the x-axis. The reference angle is always positive and less than \(\pi/2\) radians or 90 degrees.

• To find a reference angle for an angle in radians, adjust the given angle to fall within the interval \([0, 2\pi)\).
• Subtract \(2\pi\) for positive angles or add \(2\pi\) for negative angles to bring them within this range.

In our case of \(-\frac{7\pi}{4}\), adding \(2\pi\) makes it equivalent to \(\frac{\pi}{4}\), which is within the first quadrant, demonstrating how to find the reference angle, \(\frac{\pi}{4}\), easily.

Quadrants in Trigonometry

A circle is divided into four quadrants that help us understand the signs of trigonometric functions at different angles:

• The first quadrant includes angles from \(0\) to \(\frac{\pi}{2}\), with all trigonometric functions positive.
• The second quadrant spans \(\frac{\pi}{2}\) to \(\pi\), where sine is positive, but cosine and tangent are negative.
• The third quadrant is from \(\pi\) to \(\frac{3\pi}{2}\), where tangent is positive, and sine and cosine are negative.
• The fourth quadrant goes from \(\frac{3\pi}{2}\) to \(2\pi\), and cosine is positive, but sine and tangent are negative.

For \(-\frac{7\pi}{4}\), its adjusted angle \(\frac{\pi}{4}\) lies in the first quadrant, where all trigonometric functions are positive, making it easier to determine the sign of these functions.

Sine and Cosine Values

Sine and cosine are fundamental trigonometric functions that relate to the coordinates of angles on the unit circle.

• The sine function \(\sin(\theta)\) gives the y-coordinate of the point on the unit circle, which corresponds to the angle \(\theta\).
• The cosine function \(\cos(\theta)\) provides the x-coordinate of that point.

For common angles like \(\frac{\pi}{4}\), the sine and cosine values are the same: \(\frac{\sqrt{2}}{2}\). These values illustrate the symmetry of the unit circle, where \(\sin\) and \(\cos\) are equal at this angle, simplifying calculations for these specific radian measures. Understanding these core values also helps when calculating other trigonometric functions like tangent, cotangent, secant, and cosecant.