Problem 27 - Algebra 2

Question

Find the exact values of the cosine and sine of each angle. Then find the decimal values. Round your answers to the nearest hundredth. $$ -30^{\circ} $$

Step-by-Step Solution

Answer

The exact values of cosine and sine for -30 degrees are \(\sqrt{3}/2\) and \(-1/2\) respectively. The decimal approximations rounded to the nearest hundredth are 0.87 and -0.50.

Step 1: Determine the Quadrant and Referent Angle

The angle -30 degrees lies in the fourth quadrant where cosine is positive and sine is negative. Therefore, we have to use these signs when calculating these values. The referent (related acute) angle for -30 degrees is 30 degrees.

Step 2: Find Exact Values

We use the well-known exact trigonometric values for 30 degrees from the special right triangle or unit circle. The cosine of 30 degrees is \(\sqrt{3}/2\) and the sine of 30 degrees is \(1/2\). According to the sign rule from step 1, we have to take cosine as positive and sine as negative.

Step 3: Decimal Approximation

Finally, we find the decimal approximations for these values and round them to the nearest hundredth. The decimal approximation of \(\sqrt{3}/2\) is approximately 0.87 and of \(1/2\) is 0.50. So, the cosine of -30 degrees is approximately 0.87 and the sine is approximately -0.50.

Key Concepts

Exact Trigonometric Values

Trigonometric functions like sine, cosine, and tangent have specific values for certain angles. These are known as "exact trigonometric values." They are handy when calculating angles such as 30°, 45°, or 60°, and their complementary angles in different quadrants.
To find these exact values, you can often rely on special triangles or the unit circle.

• For 30°, the sine value is \( \frac{1}{2} \) and the cosine value is \( \frac{\sqrt{3}}{2} \).
• For 45°, both sine and cosine have the value \( \frac{\sqrt{2}}{2} \).
• For 60°, the sine value is \( \frac{\sqrt{3}}{2} \) and the cosine is \( \frac{1}{2} \).

You will often need to adapt these values based on the quadrant the angle resides in. Remember the signs change according to whether the sine or cosine is positive or negative in that particular quadrant.

Unit Circle

The unit circle is a crucial tool in trigonometry. It helps in visualizing the values of trigonometric functions. The unit circle is simply a circle with a radius of 1 centered at the origin of a coordinate plane. Any point on this circle can be expressed in terms of \( (\cos(\theta), \sin(\theta)) \).
The angle θ is measured from the positive x-axis, going counterclockwise.

• For an angle of 30°, its coordinates are \( \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right) \).
• These coordinates change as you move around the circle, reflecting different angle values and their quadrants.
• With negative angles, such as -30°, the movement goes clockwise but retains the same unit circle principles.

The beauty of the unit circle is how it encodes the relationships of angles and their corresponding sine and cosine values seamlessly in one simple geometric figure.

Quadrant Analysis

Understanding which quadrant an angle lies in is vital because it determines the signs of the trigonometric functions. There are four quadrants in total. Each has its own rules regarding the positivity or negativity of sine and cosine functions:

• First Quadrant (0° to 90°): Both sine and cosine are positive.
• Second Quadrant (90° to 180°): Sine is positive, cosine is negative.
• Third Quadrant (180° to 270°): Both sine and cosine are negative.
• Fourth Quadrant (270° to 360° or -0° to -90°): Sine is negative and cosine is positive.

For the angle -30°, which lies in the fourth quadrant, use these rules to adjust the exact values correctly. Cosine remains positive, while sine turns negative, giving us specific clear rules based on position and direction on the circle. This systematic approach will allow you to properly decipher trigonometric values, no matter the angle.