Problem 60 - Precalculus: A Concise Course

Question

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$ \frac{3 \pi}{4} $$

Step-by-Step Solution

Answer

The values are as follows: \(\sin(\frac{3 \pi}{4}) = \sqrt{2}/2 , \cos(\frac{3 \pi}{4}) = -\sqrt{2}/2 , \tan(\frac{3 \pi}{4}) = -1\)

Step 1: Evaluation of Sine

Evaluate the sine of the angle by using the unit circle. In the unit circle, \(\sin(\frac{3 \pi}{4})\) is equal to \(\sin(\pi - \frac{\pi}{4})\) that is equal to \(\sin(180° - 45°)\), which leads to \(\sin(135°)\). From special angles in the unit circle it is known that \(\sin(135°)\) is \(\sqrt{2}/2\)

Step 2: Evaluation of Cosine

Evaluate the cosine of the angle by using the unit circle also. In the unit circle, \(\cos(\frac{3 \pi}{4})\) is equal to \(\cos(\pi - \frac{\pi}{4})\) equals to \(\cos(180° - 45°)\) which leads to \(\cos(135°)\). From operation of special angles in the unit circle it is known that \(\cos(135°)\) is \(-\sqrt{2}/2\) as cosine is negative in the second quadrant.

Step 3: Evaluation of Tangent

Evaluate the tangent of the angle by using the ratio of sine to cosine. \(\tan(\frac{3 \pi}{4})\) is \(\sin(\frac{3 \pi}{4}) / \cos(\frac{3 \pi}{4})\). Thus, by substituting the earlier derived values, \(\tan(\frac{3 \pi}{4})\) equates to \((\sqrt{2}/2) / ( - \sqrt{2}/2)\), which simplifies to \(-1\)

Key Concepts

Unit Circle

The unit circle is a fundamental tool in trigonometry that helps us understand the relationships of sine, cosine, and tangent for all angles. It is a circle with a radius of 1, centered at the origin of the coordinate plane (0,0). Every point on the unit circle represents an angle, and the coordinates of that point give the values of cosine and sine for that angle.

• The x-coordinate of a point on the unit circle is the cosine of the angle.
• The y-coordinate is the sine of the angle.

For example, for the angle \(\frac{3\pi}{4}\), you would find this point in the second quadrant of the unit circle, where the sine is positive, and the cosine is negative. Understanding the unit circle allows you to quickly determine these values without a calculator, as seen with the \(\frac{3\pi}{4}\) angle in the example.

Special Angles

Special angles are those angles that have well-known sine, cosine, and tangent values. These angles usually include

• 0° (0 radians)
• 30° (\(\frac{\pi}{6}\))
• 45° (\(\frac{\pi}{4}\))
• 60° (\(\frac{\pi}{3}\))
• 90° (\(\frac{\pi}{2}\))

They also include their equivalents in other quadrants, like 180° and 270°.When evaluating \(rac{3 \pi}{4}\), you are essentially looking at the special angle 45° but in the second quadrant.In this quadrant:

• Sine is still positive since it corresponds to the y-coordinate, which is above the x-axis.
• Cosine is negative because the x-coordinate is to the left of the y-axis.

These known values simplify to \(rac{3 \pi}{4}\) having sine \( \frac{\sqrt{2}}{2} \) and cosine \(-\frac{\sqrt{2}}{2} \).

Tangent Ratio

The tangent of an angle is another important trigonometric function, closely related to sine and cosine. The tangent ratio is defined as the sine of the angle divided by the cosine of the angle:\[\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\]For \(rac{3\pi}{4}\), calculating the tangent is straightforward once you know sine and cosine:

• Sine is \(\frac{\sqrt{2}}{2}\)
• Cosine is \(-\frac{\sqrt{2}}{2}\)

Plug these values into the tangent formula:\[\tan\left(\frac{3\pi}{4}\right) = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}}\]The square root terms cancel each other out, leaving -1. Thus, \(\tan\left(\frac{3\pi}{4}\right) = -1\). This reflects the fact that in the second quadrant, the tangent of an angle is negative.