Problem 31 - Algebra 2

Question

For each angle \(\theta,\) find the values of \(\cos \theta\) and \(\sin \theta .\) Round your answers to the nearest hundredth. $$ -45^{\circ} $$

Step-by-Step Solution

Answer

The cosine and sine values for \(-45^{\circ}\) are 0.71 and -0.71 respectively.

Step 1: Convert angle to radians

Since trigonometric functions generally use radians, it's easier to work with this format. Convert \(-45^{\circ}\) to radians. The formula to convert degrees to radians is \(\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}\). Therefore, we have \(-45^{\circ} \times \frac{\pi}{180} = -\frac{\pi}{4}\) rad.

Step 2: Compute cosine

Use the unit circle to compute the cosine value for \(-\frac{\pi}{4}\) rad. When we plug this value into the cosine function, we get \(\cos(-\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\). This value when rounded to hundredths is 0.71.

Step 3: Compute sine

Use the unit circle to compute the sine value for angle \(-\frac{\pi}{4}\) rad. The sine function gives us \(\sin(-\frac{\pi}{4}) = -\frac{\sqrt{2}}{2}\), which when rounded to hundredths is -0.71

Key Concepts

Understanding Cosine

When discussing trigonometric functions, cosine is one of the primary functions you'll encounter. The cosine of an angle in a right triangle, specifically, is the ratio of the adjacent side to the hypotenuse.

In the unit circle, the cosine of an angle represents the x-coordinate of a point on the circle. Since the unit circle has a radius of 1, this makes calculations easy. For a given angle \(-\frac{\pi}{4}\) (which is equivalent to \(-45^\circ\)), the cosine value is \(\cos(-\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\).This value, when rounded to the nearest hundredth, becomes 0.71. In the unit circle, this value demonstrates how far along the x-axis the point corresponding to \(-\frac{\pi}{4}\) radians is. Cosine values vary from -1 to 1, and they repeat every \(2\pi\) radians, a property known as periodicity.Cosine is commonly abbreviated as "cos," making it easy to refer to this key trigonometric function.

The Role of Sine

Sine is another crucial trigonometric function that complements cosine. In the context of a right triangle, the sine of an angle is the ratio of the opposite side to the hypotenuse. However, when using the unit circle, sine reflects the y-coordinate of a point at a specific angle on the circumference.

For the angle \(-\frac{\pi}{4}\), the sine value is \(\sin(-\frac{\pi}{4}) = -\frac{\sqrt{2}}{2}\). When rounded to the nearest hundredth, this becomes -0.71. This indicates the position along the y-axis for the point representing \(-\frac{\pi}{4}\) radians.Just like cosine, sine values also range from -1 to 1, and the function is periodic with a cycle of \(2\pi\) radians. It helps to keep in mind that sine is abbreviated as "sin," allowing for quick reference when solving problems involving angles.

Exploring the Unit Circle

The unit circle is a powerful tool in trigonometry that helps visualize and solve problems involving trigonometric functions like sine and cosine. This circle is defined as having a radius of 1 and is centered at the origin of a coordinate plane.

Within this circle, any angle can be represented as a point where the terminal side of the angle intersects the circle. The coordinates of this point are given as \((\cos \theta, \sin \theta)\). Thus, the unit circle makes it easy to quickly identify the cosine and sine of any angle.For example, with \(-45^\circ\) or \(-\frac{\pi}{4}\) radians, the cosine and sine values can be directly observed from the x and y coordinates of the corresponding point. The unit circle not only simplifies calculations, it also helps illustrate the periodic and symmetrical properties of trigonometric functions, revealing patterns you can use in broader mathematical contexts.