Problem 67 - Algebra 2

Question

What is the exact value of \(\cos (-210)^{\circ} ?\) Show your work.

Step-by-Step Solution

Answer

\(\cos (-210^{\circ}) = -\sqrt{3} / 2\).

Step 1: Transforming the Negative Angle to Positive

Since cosine has a cycle of \(360^{\circ}\), we can add this to our negative angle to get an equivalent positive angle. Thus, -210 degrees is equivalent to -210 + 360 = 150 degrees. So, \(\cos (-210^{\circ}) = \cos (150^{\circ})\).

Step 2: Computing the Cosine of the Angle

150 degrees is in the second quadrant, where cosine is negative. Since \(\cos (30^{\circ}) = \sqrt{3} / 2\) and \(150^{\circ} = 180^{\circ} - 30^{\circ}\), we should have \( \cos (150^{\circ}) = -\cos (30^{\circ}).\)

Step 3: Final Result

Substituting the value of \(\cos (30^{\circ})\) from the above step, our final result is: \(\cos (150^{\circ}) = -\sqrt{3} / 2\). Thus, \(\cos (-210^{\circ}) = -\sqrt{3} / 2\).

Key Concepts

Cosine function

The cosine function is one of the fundamental trigonometric functions. It describes the relationship between the angle formed in a right triangle and the adjacent and hypotenuse sides of this triangle. The cosine of an angle, generally denoted by \( \cos \theta \), essentially answers how much of the length from the hypotenuse projects onto the adjacent side.When dealing with angles, these can be positive or negative, and it’s useful to know that the cosine function is periodic. This means it repeats its values in a regular pattern every 360 degrees. For instance, \( \cos \theta \) is equal to \( \cos (\theta + 360^{\circ}k) \) for any integer \( k \). Moreover, the cosine function is **even**, which implies symmetry along the y-axis. Therefore, \( \cos(-\theta) = \cos(\theta) \). This property helps when calculating cosine values for negative angles, simplifying them to their positive equivalents.

Angle transformations

Angle transformations are key to simplifying trigonometric problems. This involves converting angles to different but equivalent angles that are easier to evaluate.**Negative Angles and Positive Equivalents**
Negative angles can be tricky, but with cosine's periodicity of 360 degrees, it's possible to find a positive equivalent by adding 360 degrees to the negative angle. For example, to convert \(-210^{\circ}\) into a positive angle, we can compute it as:

• \(-210^{\circ} + 360^{\circ} = 150^{\circ}\)

By transforming \(-210^{\circ}\) to \(150^{\circ}\), we make the angle's trigonometric function easier to handle.**Reference Angles**
Finding the reference angle also helps in computations. The reference angle is formed with the x-axis and is always acute (less than 90 degrees). For \(150^{\circ}\), the reference angle is \(30^{\circ}\) since it can be calculated by \(180^{\circ} - 150^{\circ}\). This reference angle makes it easier to use known trigonometric values.

Quadrant angles

The value of trigonometric functions, like cosine, changes based on the quadrant where the angle lies. Understanding this can help accurately determine the sign and value of these functions.**Quadrants Overview**
In the coordinate plane, there are four quadrants:

• **Quadrant I**: All trigonometric functions are positive.
• **Quadrant II**: Only sine and cosecant are positive, while cosine and tangent are negative.
• **Quadrant III**: Only tangent and cotangent are positive.
• **Quadrant IV**: Only cosine and secant are positive.

Knowing this sign change is pivotal. Since \(150^{\circ}\) is in Quadrant II, \(\cos(150^{\circ})\) is negative. **Applying Knowledge**
When solving \(\cos(150^{\circ})\), realizing it is in the second quadrant lets us know that cosine is negative here. Since \(150^{\circ} = 180^{\circ} - 30^{\circ} \), it uses the reference angle \(30^{\circ}\), but its cosine value is negative, thus \( \cos(150^{\circ}) = -\cos(30^{\circ}) \). This understanding helps simplify and solve trigonometric expressions quickly.