Problem 19 - Contemporary Precalculus

Question

Simplify the given expression. $$\sin 3 \cos 5-\cos 3 \sin 5$$

Step-by-Step Solution

Answer

Question: Simplify the trigonometric expression: $$\sin 3 \cos 5 - \cos 3 \sin 5$$ Answer: $$\sin(-2)$$

Step 1: Identify a and b from the given expression

Here, we can see that a = 3 and b = 5.

Step 2: Apply the sine angle addition formula

Using the angle addition formula for sine, we have: $$\sin(a - b) = \sin a \cos b - \cos a \sin b$$ Substitute a = 3 and b = 5 into the formula: $$\sin(3 - 5) = \sin 3 \cos 5 - \cos 3 \sin 5$$

Step 3: Simplify the expression

Now, we can simplify the expression by substituting the value of a - b: $$\sin(-2) = \sin 3 \cos 5 - \cos 3 \sin 5$$ Therefore, the simplified expression is: $$\sin(-2)$$

Key Concepts

Angle Addition Formula

One of the powerful tools in trigonometry is the angle addition formula. It allows us to find the trigonometric functions of the sum or difference of two angles easily. In this case, we used the angle addition formula for sine:\[ \sin(a - b) = \sin a \cos b - \cos a \sin b \]This formula shows how the sine function of a difference of angles can be broken down into a combination of sine and cosine functions of individual angles. Here, the original expression \( \sin 3 \cos 5 - \cos 3 \sin 5 \) perfectly fits this pattern. It can be transformed using the formula, making it a powerful method to simplify expressions in trigonometry.

Sine Function

The sine function is one of the primary trigonometric functions. It is essential in understanding waves, oscillations, and circles. The function takes an angle and returns the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.Key properties of the sine function include:

• The range is between -1 and 1.
• It is periodic with a period of \(2\pi\).
• The function is odd, meaning \(\sin(-x) = -\sin(x)\).

For odd functions like sine, the negative angle theorem \( \sin(-x) = -\sin(x) \) helps in simplifying or evaluating expressions involving negative angles. This property was utilized in the final expression \(\sin(-2)\). This characteristic of sine functions is integral for solving various trigonometric problems and is evident in the exercise solution.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions can be daunting, but understanding the core strategies can make the process easier. The exercise demonstrates a typical simplification where an initially complex expression is reduced using well-known identities.Here are some tips for simplifying such expressions:

• Identify any potential trigonometric identities that match the expression.
• Substitute the matched identities to rewrite the expression in a simpler form.
• Simplification often involves recognizing patterns that fit into rules, like the angle addition formula.

In the given exercise, the expression \( \sin 3 \cos 5 - \cos 3 \sin 5 \) was recognized as an instance of the angle addition formula. This allowed us to simplify it directly into \( \sin(-2) \). Always be on the lookout for such opportunities to apply identities, as they can transform complex expressions into something manageable. It's about breaking it down, understanding the relationships, and converting it into a simpler, equivalent form.