Problem 24 - Contemporary Precalculus

Question

In Exercises \(15-29,\) find the exact value of the sine, cosine, and tangent of the number, without using a calculator. $$11 \pi / 6$$

Step-by-Step Solution

Answer

Answer: The exact values for the angle $$11 \pi / 6$$ are: $$\sin\left(\frac{11\pi}{6}\right)=-\frac{1}{2}$$ $$\cos\left(\frac{11\pi}{6}\right)=\frac{\sqrt{3}}{2}$$ $$\tan\left(\frac{11\pi}{6}\right)=-\frac{1}{\sqrt{3}}$$

Step 1: Convert the given angle to an equivalent angle between \(0\) and \(2\pi\)

Notice that $$11 \pi / 6$$ is more than one full rotation of the unit circle, which is \(2\pi\). To find the equivalent angle between \(0\) and \(2\pi\), we can subtract the multiple of \(2\pi\): $$\frac{11\pi}{6} - 2\pi = \frac{11\pi}{6} - \frac{12\pi}{6} = -\frac{1\pi}{6}$$ Since the angle is negative, we need to add \(2\pi\) to find the equivalent positive angle: $$-\frac{1\pi}{6} + 2\pi = -\frac{1\pi}{6} + \frac{12\pi}{6} = \frac{11\pi}{6}$$

Step 2: Determine the reference angle

Reference angles are always measured with respect to the x-axis. Now, we have $$11 \pi/6$$ in the positive direction, which lies in the fourth quadrant. We subtract \(\pi\) to get the reference angle in the fourth quadrant: $$\text{Reference Angle} = \frac{11\pi}{6} - \frac{10\pi}{6} = \frac{\pi}{6}$$

Step 3: Find the sine, cosine, and tangent of the reference angle

For an angle of $$\frac{\pi}{6}$$ radians (30°), the values of sine, cosine, and tangent are well-known: $$\sin\left(\frac{\pi}{6}\right)=\frac{1}{2}$$ $$\cos\left(\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2}$$ $$\tan\left(\frac{\pi}{6}\right)=\frac{\sin\left(\frac{\pi}{6}\right)}{\cos\left(\frac{\pi}{6}\right)}=\frac{1}{\sqrt{3}}$$

Step 4: Determine the signs of the trigonometric functions in the fourth quadrant

In the fourth quadrant, sine is negative, cosine is positive, and tangent is negative. Therefore, we have: $$\sin\left(\frac{11\pi}{6}\right)=-\sin\left(\frac{\pi}{6}\right)=-\frac{1}{2}$$ $$\cos\left(\frac{11\pi}{6}\right)=\cos\left(\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2}$$ $$\tan\left(\frac{11\pi}{6}\right)=-\tan\left(\frac{\pi}{6}\right)=-\frac{1}{\sqrt{3}}$$ So, the exact values of sine, cosine, and tangent for the given angle $$11 \pi / 6$$ are: $$\sin\left(\frac{11\pi}{6}\right)=-\frac{1}{2}$$ $$\cos\left(\frac{11\pi}{6}\right)=\frac{\sqrt{3}}{2}$$ $$\tan\left(\frac{11\pi}{6}\right)=-\frac{1}{\sqrt{3}}$$

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. Using the unit circle, we can determine the trigonometric functions for various angles, as the coordinates on the circle correspond to specific values of sine and cosine.

• The x-coordinate of a point on the unit circle represents the cosine of the angle.
• The y-coordinate represents the sine of the angle.
• Angles are measured from the positive x-axis, moving counterclockwise.

The unit circle also helps in understanding how angles repeat every full rotation (i.e., every \(2\pi\) radians or 360 degrees). This is essential in finding equivalent angles within the circle.

Radians to Degrees Conversion

Converting between radians and degrees is a basic skill in trigonometry. Radians are often used because they relate directly to the unit circle.
To convert from radians to degrees, use the formula:
\[\text{degrees} = \text{radians} \times \frac{180}{\pi}\] For example, an angle of \(\frac{\pi}{6}\) radians can be converted by:\(\frac{\pi}{6} \times \frac{180}{\pi} = 30\) degrees.
Remember, multiples of \(\pi\) describe rotations on the unit circle easily. This is why trigonometry often prefers radians for calculations related to periodic functions.

Reference Angles

Reference angles simplify the process of finding the trigonometric values of any angle. They are positive angles formed between the terminal side of the given angle and the x-axis.

• Reference angles are always positive and less than \(\pi/2\) (or 90 degrees).
• For the fourth quadrant, the reference angle is found by subtracting the angle from \(2\pi\).
• Knowledge of reference angles allows us to use known values to find the trigonometric functions in any quadrant.

For our example of \(\frac{11\pi}{6}\), its reference angle is \(\frac{\pi}{6}\), which lies in the first quadrant.

Exact Trigonometric Values

Knowing exact values of trigonometric functions helps solve problems without calculators. Common exact values stem from specific angles like 30°, 45°, and 60°:

• \(\sin(30^\circ) = \sin(\frac{\pi}{6}) = \frac{1}{2}\)
• \(\cos(30^\circ) = \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}\)
• \(\tan(30^\circ) = \tan(\frac{\pi}{6}) = \frac{1}{\sqrt{3}}\)

For \(\frac{11\pi}{6}\), we apply these reference values while considering the quadrant's sign rules:

• Sine is negative in the fourth quadrant.
• Cosine remains positive.
• Tangent is also negative.

These rules allow us to determine:\( \sin\left(\frac{11\pi}{6}\right) = -\frac{1}{2}, \cos\left(\frac{11\pi}{6}\right) = \frac{\sqrt{3}}{2}, \tan\left(\frac{11\pi}{6}\right) = -\frac{1}{\sqrt{3}}\).