Problem 27 - Algebra 2

Question

Find each exact value. Use a sum or difference identity. $$ \cos 135^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \( \cos 135^{\circ} \) is \( -\sqrt{2}/2 \).

Step 1: Express the angle

Express 135 degrees as a sum or difference of angles with known cosine values. We will express 135 degrees as the sum of 90 degrees and 45 degrees: \(135^{\circ} = 90^{\circ} + 45^{\circ}\)

Step 2: Apply the sum or difference identity

Now, apply the cosine sum identity \( \cos(A + B) = \cos A \cos B - \sin A \sin B \), where A is 90 degrees and B is 45 degrees. It becomes: \( \cos 135^{\circ} = \cos(90^{\circ} + 45^{\circ}) = \cos 90^{\circ} \cos 45^{\circ} - \sin 90^{\circ} \sin 45^{\circ} \)

Step 3: Compute the values

The cosine of 90 degrees is 0, the cosine of 45 degrees is \( \sqrt{2}/2 \), the sine of 90 degrees is 1, and the sine of 45 degrees is \( \sqrt{2}/2 \). Substitute these values: \( \cos 135^{\circ} = 0 * \sqrt{2}/2 - 1 * \sqrt{2}/2 = -\sqrt{2}/2 \)

Key Concepts

Sum and Difference Identities

To find exact trigonometric values, especially for angles like 135 degrees, we use sum and difference identities. These identities allow us to express the trigonometric function of a sum or a difference of angles in terms of the functions of individual angles. In our example, we know the identity for cosine, which is:

• \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)
• \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)

To effectively use these identities, choose angles A and B that are easily computable using standard trigonometric values. For 135 degrees, an effective choice is 90 degrees and 45 degrees. This transforms our problem into a calculation that can be done with known values of these standard angles. Every step follows logically when you break down complex angles into simpler constituent parts, making use of these identities.

Cosine Function

The cosine function is fundamental in trigonometry. It is one of the primary trigonometric functions and finds its basis in the unit circle. For an angle \( \theta \) in a right triangle, and the angle is taken from the unit circle, cosine represents the length of the adjacent side divided by the hypotenuse.In terms of angles, it gives the x-coordinate of the point on the unit circle. This means for any angle \( \theta \), \( \cos \theta \) provides the horizontal distance from the origin, showing us how much it lies from the vertical line through the origin.Characteristics of the cosine function include:

• It is an even function, meaning \( \cos(-\theta) = \cos \theta \).
• The values oscillate between -1 and 1.
• It has a period of \(360^{\circ}\) or \(2\pi\) radians.

Understanding the behavior of cosine aids in applying these identities to find precise angle values easily, as seen in calculating \( \cos 135^{\circ} \).

Exact Trigonometric Values

Having knowledge of exact trigonometric values helps tremendously with calculations. They are often derived from special right triangles, such as the 45-45-90 triangle and the 30-60-90 triangle, or the unit circle.Here are some key values:

• \( \cos 45^{\circ} = \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \sin 90^{\circ} = 1, \cos 90^{\circ} = 0 \)
• \( \sin 30^{\circ} = \frac{1}{2}, \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)
• \( \sin 60^{\circ} = \frac{\sqrt{3}}{2}, \cos 60^{\circ} = \frac{1}{2} \)

By using these values, we can calculate more complex expressions without needing a calculator. For example, to find \( \cos 135^{\circ} \), substitute these known values into the sum identity expression, arriving at \( \cos 135^{\circ} = -\frac{\sqrt{2}}{2} \). This highlights how knowing exact trigonometric values can simplify our computations in solving trigonometric problems efficiently.