Problem 32 - Contemporary Precalculus

Question

In Exercises \(31-36\), write the expression as a single real mum. ber. Do not use decimal approximations. IHint: Exercises \(15-21 \text { may be helpful. }]\) $$\sin (\pi / 6) \cos (\pi / 2)-\cos (\pi / 6) \sin (\pi / 2)$$

Step-by-Step Solution

Answer

Question: Simplify the given trigonometric expression: $$\sin (\pi / 6) \cos (\pi / 2)-\cos (\pi / 6) \sin (\pi / 2)$$ Answer: The simplified trigonometric expression is equal to \(-\frac{\sqrt{3}}{2}\).

Step 1: Recall angle sum formula for sine function

We start by recalling the angle sum formula for the sine function: $$\sin(a - b) = \sin a \cos b - \cos a \sin b$$ This formula allows us to find the sine of the difference of two angles a and b in terms of the sines and cosines of a and b.

Step 2: Identify the angle difference in the given expression

We can see that the given expression has the form of our angle sum formula: $$\sin (\pi / 6) \cos (\pi / 2)-\cos (\pi / 6) \sin (\pi / 2)$$ Comparing this to our angle sum formula, we can observe that: a = \(\pi / 6\), and b = \(\pi / 2\)

Step 3: Calculate the difference of the angles

Our goal is to find the angle difference (\(a - b\)). To do this, we can use the values of a and b that we found in step 2: $$a - b = \frac{\pi}{6} - \frac{\pi}{2}$$ Now, find a common denominator and subtract the fractions: $$a - b = \frac{\pi}{6} - \frac{3\pi}{6} = -\frac{2\pi}{6} = -\frac{\pi}{3}$$

Step 4: Use the angle difference to simplify the expression

Now that we have found the angle difference, we can use it to simplify the given expression, which matches our angle sum formula for the sine function: $$\sin(\frac{\pi}{6}\cos(\frac{\pi}{2})-\cos(\frac{\pi}{6}\sin(\frac{3\pi}{2}$$ $$= \sin(-\frac{\pi}{3})$$

Step 5: Find the value of the sine function for the angle difference

We know the value of the sine function for some common angles, including the angle \(\pi / 3\). Since \(-\pi / 3\) is the negative of a common angle, we can use the fact that sine function is odd: $$\sin(-x) = -\sin x$$ So, we get: $$\sin(-\frac{\pi}{3}) = -\sin(\frac{\pi}{3})$$

Step 6: Final answer

Using the fact that \(\sin(\pi / 3) = \frac{\sqrt{3}}{2}\), we can now find our final answer: $$-\sin(\frac{\pi}{3}) = -\frac{\sqrt{3}}{2}$$

Key Concepts

Angle Sum Formula

The angle sum formula for the sine function is an essential tool in trigonometry. Specifically, it states that for two angles \( a \) and \( b \):\[\sin(a - b) = \sin a \cos b - \cos a \sin b\]This formula helps us express the sine of a difference between two angles in terms of the sine and cosine values of the individual angles. This can be especially useful when you need to evaluate expressions without using a calculator.In the given exercise, recognizing the structure of this formula allows us to transform a seemingly complex expression into something more manageable. Always remember this formula as it appears frequently in various trigonometric problems:- It can simplify complex expressions.- It's helpful in problems involving angle manipulation.It's a relationship that ties the geometry of triangles to the algebra of trigonometric identities.

Sine Function

The sine function is one of the fundamental functions in trigonometry. It describes the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right triangle.The sine function is defined for all real numbers and is periodic with a period of \( 2\pi \). Key properties include:- **Odd Function:** The sine function satisfies \( \sin(-x) = -\sin x \). This property was utilized in the exercise to determine \( \sin(-\frac{\pi}{3}) \).- **Value Range:** \( \sin x \) will always lie between \(-1\) and \(1\).In trigonometry, certain angles have known sine values:- \( \sin \left( \frac{\pi}{6} \right) = \frac{1}{2} \)- \( \sin \left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2} \)By understanding these properties, you can evaluate and manipulate trigonometric expressions more easily.

Angle Difference

The concept of angle difference plays a significant role in this problem. When you have two angles and find their difference, you can use trigonometric identities like the angle sum formula to simplify expressions.For instance, in the exercise:- Given angles were \( a = \frac{\pi}{6} \) and \( b = \frac{\pi}{2} \).- The difference was calculated as \( a - b = -\frac{\pi}{3} \).This value represents a specific angle for which the sine function can be evaluated using known identities. Calculating angle differences helps in simplifying trigonometric expressions:- Convert complicated expressions into single trigonometric functions.- Use known trigonometric values for simplification.By understanding how to find and use angle differences, solving trigonometric problems becomes more straightforward and logical.