Problem 15
Question
Exer. \(11-16\) : Express as a trigonometric function of one angle. $$\cos 3 \sin (-2)-\cos 2 \sin 3$$
Step-by-Step Solution
Verified Answer
The expression is \( \sin 5 \).
1Step 1: Identify the Expression
The given expression is \( \cos 3 \sin (-2) - \cos 2 \sin 3 \). We need to express this in terms of a single trigonometric function.
2Step 2: Apply Sine Angle Sum Identity
Recall the sine angle sum identity: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \). We can rewrite the expression as a single sine function using the identity by recognizing the structure of \( -\cos 3 \sin 2 \) as \(-\sin(A+B) = -\sin(3-2)\).
3Step 3: Simplify Using Sine Angle Sum Identity
By substituting into the identity: \[-(\cos 3 \sin (-2) - \cos 2 \sin 3) = -\sin(3 + 2) = -\sin 5\].
4Step 4: Combine and Rewrite the Expression
Since the expression is negative, applying the negative as part of the identity transformation yields:\[ \cos 3 \sin (-2) - \cos 2 \sin 3 = \sin 5 \].
Key Concepts
Sine Angle Sum IdentityTrigonometric FunctionsSimplifying Trigonometric Expressions
Sine Angle Sum Identity
The Sine Angle Sum Identity is an important trigonometric identity that helps us rewrite expressions involving angles added together. This identity states that for any angles \(A\) and \(B\), \(\sin(A + B) = \sin A \cos B + \cos A \sin B\). This means that we can express the sine of a sum of two angles as a sum of products of sines and cosines of those angles.
Understanding this identity is key to solving problems where we need to express more complex trigonometric expressions as simpler single-angle functions. By recognizing parts of an expression that fit the pattern of this identity, we can simplify and transform the expression more easily. In our exercise, we used the Sine Angle Sum Identity to rewrite an expression involving multiple angles into a simpler form, which made it easier to combine and express it as \(\sin 5\).
Understanding this identity is key to solving problems where we need to express more complex trigonometric expressions as simpler single-angle functions. By recognizing parts of an expression that fit the pattern of this identity, we can simplify and transform the expression more easily. In our exercise, we used the Sine Angle Sum Identity to rewrite an expression involving multiple angles into a simpler form, which made it easier to combine and express it as \(\sin 5\).
Trigonometric Functions
Trigonometric functions are fundamental to understanding relationships in geometry, especially involving right triangles and the unit circle. The basic trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)), each of which relates an angle of a right triangle to ratios of its sides.
In the context of angle sum identities, sine and cosine are particularly important. These functions help us quantify and compute angles and their associations with specific lengths on triangles. By using the properties of these functions, such as symmetry and periodicity, we can transform and simplify complex expressions, making calculations easier and helping us to draw meaningful geometric or mathematical conclusions.
In the context of angle sum identities, sine and cosine are particularly important. These functions help us quantify and compute angles and their associations with specific lengths on triangles. By using the properties of these functions, such as symmetry and periodicity, we can transform and simplify complex expressions, making calculations easier and helping us to draw meaningful geometric or mathematical conclusions.
- \(\sin\) corresponds to the ratio of the length of the opposite side to the hypotenuse.
- \(\cos\) corresponds to the ratio of the length of the adjacent side to the hypotenuse.
- These functions are cyclical, with periodic wave-like behaviors that are both predictable and have useful applications in various fields like engineering, physics, and computer science.
Simplifying Trigonometric Expressions
Simplifying trigonometric expressions is a process of manipulating and rewriting them to achieve a simpler or more elegant form. This often involves using trigonometric identities, like the sine angle sum identity, to combine or eliminate terms. The goal is to reduce complexity, sometimes by reducing the number of terms or by expressing the function in terms of a single angle.
In the given solution, our task was to take a more complex expression \(\cos 3 \sin (-2) - \cos 2 \sin 3\) and simplify it using the sine angle sum identity. By identifying applicable identities, recognizing patterns, and substituting appropriately, the expression was transformed into \(\sin 5\).
In the given solution, our task was to take a more complex expression \(\cos 3 \sin (-2) - \cos 2 \sin 3\) and simplify it using the sine angle sum identity. By identifying applicable identities, recognizing patterns, and substituting appropriately, the expression was transformed into \(\sin 5\).
- Recognize common identities: Identifying known patterns can help to see where transformations can simplify the expression.
- Careful algebraic manipulation: Pay attention to negative signs and arithmetic operations to ensure correct simplification.
- Reduction to simple forms: Aim to achieve a single trigonometric function that reflects the combined action represented by the original expression.
Other exercises in this chapter
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