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Chapter 3: Problem 2

Use the unit circle to evaluate each function. \(\cos 225^{\circ}\)

Short Answer

Expert verified
\(\cos 225^{\circ} = -\frac{\sqrt{2}}{2}\)

Step by step solution

01

Identify the Angle on the Unit Circle

Locate the angle \(225^{\circ}\) on the unit circle. This angle is in the third quadrant because it is between \(180^{\circ}\) and \(270^{\circ}\).
02

Determine the Reference Angle

The reference angle for \(225^{\circ}\) is \(225^{\circ} - 180^{\circ} = 45^{\circ}\). Reference angles help in identifying the coordinates on the unit circle.
03

Recall Coordinates for the Reference Angle

For the reference angle \(45^{\circ}\), in the first quadrant, the coordinates are \((\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\).
04

Adjust Coordinates for the Third Quadrant

In the third quadrant, both the x and y coordinates are negative. Hence, transform the coordinates to \((-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})\).
05

Find the Cosine Value

The cosine value is the x-coordinate of the point on the unit circle. Therefore, \(\cos 225^{\circ} = -\frac{\sqrt{2}}{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Function
The cosine function is one of the primary trigonometric functions and plays a vital role in understanding angles and triangles, especially in the context of a unit circle. It associates an angle with the x-coordinate of a point on the unit circle.
  • A unit circle is a circle with a radius of one centered at the origin (0,0) of a coordinate plane.
  • The cosine of an angle provides the horizontal distance from the center of the circle to its boundary for that angle.
  • This function is periodic, repeating values at regular intervals of 360掳 or 2蟺 radians.
In our example of evaluating \(\cos 225^{\circ}\), the angle lies in the third quadrant of the unit circle. Calculating the cosine involves identifying the x-coordinate of the corresponding point. Here, because 225掳 is in the third quadrant where both coordinates are negative, \(-\frac{\sqrt{2}}{2}\) is the cosine value.
Reference Angle
A reference angle is critical in simplifying the process of finding trigonometric values for non-standard angles. It is the acute angle formed by the terminal side of a given angle and the horizontal axis.
  • The reference angle always lies between 0掳 and 90掳 (or between 0 and 蟺/2 radians).
  • This angle helps to utilize known values of trigonometric functions from the first quadrant.
  • The rule to find the reference angle varies slightly depending on the angle's quadrant.
For example, the reference angle for \(225^{\circ}\) is \(45^{\circ}\) because \(225^{\circ} - 180^{\circ} = 45^{\circ}\). With this, you can leverage the cosine values from the first quadrant and adjust the sign as per the quadrant of the original angle. Here, this leads us to \(-\frac{\sqrt{2}}{2}\) for \(\cos 225^{\circ}\).
Trigonometric Functions
Trigonometric functions, including cosine, sine, and tangent, are fundamental to mathematics and have a widespread application across different fields. These functions relate the angles of a triangle to the lengths of its sides in right-angled triangles.
  • The major trigonometric functions are sine, cosine and tangent, each representing a ratio of two sides of a triangle.
  • Cosine (cos) relates the adjacent side over the hypotenuse.
  • Sine (sin) relates the opposite side over the hypotenuse.
  • Tangent (tan) relates the opposite side over the adjacent side.
For the unit circle, these functions record the coordinates of the endpoints of the radius and provide a way to calculate angular amplitudes in cycles or oscillations. While \(\cos 225^{\circ} = -\frac{\sqrt{2}}{2}\), the sine and tangent values can also illustrate similar methods of calculation through the trigonometric rules associated with reference angles and unit circle properties.

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Most popular questions from this chapter

When Lance Armstrong blazed up Mount Ventoux in the 2002 Tour, he was equipped with a 150 -millimeter-diameter chainring and a 95 -millimeterdiameter sprocket. Lance is known for maintaining a very high cadence, or pedal rate. If he was pedaling at a rate of 90 revolutions per minute, find his speed in kilometers per hour. \((1 \mathrm{~km}=1,000,000 \mathrm{~mm})\)

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Use a calculator to find \(\theta\) to the nearest tenth of a degree, if \(0^{\circ}<\theta<360^{\circ}\) and $$ \sec \theta=-1.7876 \text { with } \theta \text { in QIII } $$

Use a calculator to find the following. $$ \cos \left(-315^{\circ}\right) $$
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