Problem 33 - Algebra 2

Question

For each angle \(\theta,\) find the values of \(\cos \theta\) and \(\sin \theta .\) Round your answers to the nearest hundredth. $$ -95^{\circ} $$

Step-by-Step Solution

Answer

After going through these steps, you should get the cosine and sine values of -95 degrees. They are approximate values and should be rounded to the nearest hundredth.

Step 1: Convert Degrees to Radians

Firstly, convert the angle from degrees into radians since the standard unit in trigonometry is radian. The conversion formula from degrees to radians is \(\theta = \frac{\pi}{180} \times \text{angle in degrees}\). Thus, we compute, \(\theta = \frac{\pi}{180} \times -95\).

Step 2: Find the Equivalent Positive Angle

Since trigonometric functions are cyclic, it is helpful to find an equivalent positive angle for the negative one provided. We can do this by adding 360 degrees (or \(2\pi\) radians) to the given negative angle. The result gives an equivalent angle in the first rotation of the unit circle. Using this logic, we add \(2\pi\) to the \(\theta\) found in step 1.

Step 3: Calculate Cosine and Sine Values

Now, we calculate the cosine and sine values for the obtained angle from step 2. The cosine value represents the x-coordinate and the sine value represents the y-coordinate of the point where the terminal side of the angle intersects the unit circle. For this, we use the built-in functions in our calculator to compute these values, taking care to ensure our calculator is in radian mode. Also, we round our answers to the nearest hundredth as the problem instructs.

Key Concepts

Angle Conversion

Angle conversion is a crucial part of trigonometry, especially when dealing with different units like degrees and radians. In most trigonometric calculations, radians are the standard units used. To convert an angle from degrees to radians, it's important to use the conversion formula:

• \(\theta = \frac{\pi}{180} \times \text{angle in degrees}\)

This equation stems from the fact that a full circle, which is 360 degrees, is equivalent to \(2\pi\) radians. So, by dividing and analyzing parts of the circle, this conversion becomes straightforward.
For example, converting \(-95^\circ\) involves multiplying \(-95\) by \(\frac{\pi}{180}\), resulting in \(-\frac{95\pi}{180}\), which simplifies further into radians. It's a simple multiplication step that makes understanding and visualizing trigonometric functions on the unit circle easier.

Unit Circle

The unit circle is a powerful and visual tool to help understand trigonometric functions. Its simplicity lies in its radius, which is always 1 unit long. A complete rotation around the unit circle is \(2\pi\) radians or 360 degrees.
When locating an angle on the unit circle, the

• Sine of the angle corresponds to the y-coordinate,
• Cosine of the angle corresponds to the x-coordinate of the point.

This is because every point on the unit circle represents the result of a trigonometric function. The repetitive cycle of angles, or symmetrical patterns, also helps in understanding the sine and cosine of negative angles.
For instance, if you start with \(-95^\circ\), by converting it to a positive angle within a single circle rotation, you arrive at an angle with the same sine and cosine values since the circle repeats every \(2\pi\) radians. This cyclic nature is what allows us to "wrap" angles multiple times around the circle while maintaining the same trigonometric function values.

Radians and Degrees

Understanding both radians and degrees is fundamental in trigonometry. They are simply two different units for measuring angles. Unlike degrees, which are more common in general use, radians provide a natural link to how angles relate to circles.

• Degrees: An entire circle is divided into 360 degrees. A right angle is 90 degrees.
• Radians: An entire circle is \(2\pi\) radians. A right angle is \(\frac{\pi}{2}\) radians.

This means that to convert from degrees to radians, you substitute the given angle into the formula \(\theta = \frac{\pi}{180} \times \text{angle in degrees}\). Conversely, converting from radians to degrees means using the formula \(\theta = \frac{180}{\pi} \times \text{angle in radians}\).
Learning these conversions is key to solving problems in trigonometry since trigonometric functions like sine and cosine are more naturally defined in terms of radian measures. So, when faced with problems that use degrees, it's often strategic to convert to radians first to streamline and simplify your calculations in the context of trigonometric functions.