Problem 33 - Contemporary Precalculus

Question

In Exercises \(31-36\), write the expression as a single real mum. ber. Do not use decimal approximations. IHint: Exercises \(15-21 \text { may be helpful. }]\) $$\cos (\pi / 2) \cos (\pi / 4)-\sin (\pi / 2) \sin (\pi / 4)$$

Step-by-Step Solution

Answer

Answer: $$-\frac{\sqrt{2}}{2}$$

Step 1: Define angle A and B

For our given expression, let's assign the values of A and B: $$A = \frac{\pi}{2}, B = \frac{\pi}{4}$$

Step 2: Use the angle subtraction formula for cosine

Now, we will apply the angle subtraction formula using the angles A and B that we defined: $$\cos(A - B) = \cos A \cos B - \sin A \sin B$$ Plug in the values of A and B: $$\cos\left(\frac{\pi}{2} - \frac{\pi}{4}\right) = \cos\left(\frac{\pi}{2}\right) \cos\left(\frac{\pi}{4}\right) - \sin\left(\frac{\pi}{2}\right) \sin\left(\frac{\pi}{4}\right)$$

Step 3: Evaluate trigonometric functions and simplify

Now we will evaluate the trigonometric functions at the given angles: $$\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$$ $$\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$$ $$\cos\left(\frac{\pi}{2}\right) = 0$$ $$\sin\left(\frac{\pi}{2}\right) = 1$$ Replace these values in our expression: $$\cos\left(\frac{\pi}{4}\right) = 0 \cdot \frac{\sqrt{2}}{2} - 1 \cdot \frac{\sqrt{2}}{2}$$ Simplify the expression to get: $$\cos\left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}$$ So, the given expression can be expressed as a single real number: $$\boxed{-\frac{\sqrt{2}}{2}}$$

Key Concepts

Angle Subtraction Formula

The angle subtraction formula is an essential identity in trigonometry that helps simplify expressions involving angles. Particularly, it is used to express the cosine of the difference between two angles. The formula can be written as:

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

This means that the cosine of an angle that is the difference between two other angles, \( A \) and \( B \), can be found by using these trigonometric functions of the constituent angles. It’s a very useful identity, allowing us to break down complex angle calculations into simpler parts that we can evaluate using basic trigonometric values. This comes in handy especially in problems where direct calculation seems cumbersome. In the given exercise, the formula is applied to find the simplified form of:

• \( \cos(\pi/2) \cos(\pi/4) - \sin(\pi/2) \sin(\pi/4) \)

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It is defined in terms of the coordinates of a circle with a radius of one (unit circle). For an angle \( \theta \), the cosine value corresponds to the x-coordinate of the point on the unit circle at that angle. Some key aspects of the cosine function include:

• The cosine of \( 0 \) is \( 1 \), which denotes complete alignment with the x-axis.
• The cosine of \( \pi/2 \) is \( 0 \), illustrating when the angle is perpendicular to the x-axis.
• It has a periodic nature, repeating its values every \( 2\pi \) radians.
• The cosine function is even, meaning \( \cos(-\theta) = \cos(\theta) \).

In the solution given, we used the cosine value at \( \pi/2 \) which is zero. This simplifies the expression greatly, as any number multiplied by zero equals zero. Understanding these basic values and how to apply them makes solving trigonometric identities much more straightforward.

Sine Function

Similar to the cosine function, the sine function is another cornerstone of trigonometry. It also relies on the unit circle but focuses on the y-coordinate of the same point. For an angle \( \theta \), the value of sine corresponds to this vertical distance. Some core properties of the sine function include:

• The sine of \( 0 \) is \( 0 \), showing alignment exactly at the origin.
• The sine of \( \pi/2 \) is \( 1 \), representing the maximum positive y-coordinate.
• It exhibits periodic behavior, cycling every \( 2\pi \) radians.
• The sine function is odd, which means \( \sin(-\theta) = -\sin(\theta) \).

In the context of solving the trigonometric identity, we needed the sine value at \( \pi/2 \) which is \( 1 \). This result is essential as it transforms the expression into a simple calculation. Using known sine values effectively aids in breaking down and solving more advanced trigonometric expressions.