Problem 29
Question
In Exercises 29 - 36, write the expression as the sine, cosine,or tangent of an angle. \( \sin 3 \cos 1.2 - \cos 3 \sin 1.2 \)
Step-by-Step Solution
Verified Answer
The written expression as the sine of an angle is \( \sin(1.8) \).
1Step 1: Identify the trigonometric formula
Identify that the given expression matches the formula \( \sin(a - b) = \sin a \cos b - \cos a \sin b \). This is the difference of angles formula.
2Step 2: Substitute a and b in the formula
Substitute \( a = 3 \) and \( b = 1.2 \) in the formula. So, the given expression is equal to \( \sin(3 - 1.2) \).
Key Concepts
Trigonometric IdentitiesSine FunctionCosine FunctionTangent Function
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables involved. They are essential tools for simplifying trigonometric expressions and solving trigonometric equations. One of the fundamental identities is the difference of angles formula, which is central to our exercise.
It states that for any two angles, say 'a' and 'b', the sine of the difference between these angles can be expressed as \(\sin(a-b) = \sin a \cos b - \cos a \sin b\). This is an invaluable tool when we encounter products of sines and cosines in mathematical problems. By recognizing patterns that fit this identity, complex expressions can be rewritten as a single trigonometric function of an angle, greatly simplifying the calculation.
It states that for any two angles, say 'a' and 'b', the sine of the difference between these angles can be expressed as \(\sin(a-b) = \sin a \cos b - \cos a \sin b\). This is an invaluable tool when we encounter products of sines and cosines in mathematical problems. By recognizing patterns that fit this identity, complex expressions can be rewritten as a single trigonometric function of an angle, greatly simplifying the calculation.
Sine Function
The sine function, denoted as \(\sin\), is one of the primary trigonometric functions and relates the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle to an angle. It's a periodic function that oscillates between -1 and 1, and it's defined for all real numbers.
Understanding the sine function is crucial for exercises like ours, where the sine of an angle difference is needed. In a broader context, the sine function describes wave-like phenomena such as sound waves and light waves, making it also vital in physics and engineering.
Understanding the sine function is crucial for exercises like ours, where the sine of an angle difference is needed. In a broader context, the sine function describes wave-like phenomena such as sound waves and light waves, making it also vital in physics and engineering.
Cosine Function
The cosine function, represented by \(\cos\), is another core trigonometric function. It relates the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle to an angle. Like the sine function, it is also periodic and oscillates within the range of -1 to 1.
In our exercise, we encounter the cosine function in combination with the sine function as part of the difference of angles formula. The function is essential in various fields, including geometry, where it is used to determine the length of sides, angles in polygons, and in analyzing the properties of periodic functions.
In our exercise, we encounter the cosine function in combination with the sine function as part of the difference of angles formula. The function is essential in various fields, including geometry, where it is used to determine the length of sides, angles in polygons, and in analyzing the properties of periodic functions.
Tangent Function
The tangent function, expressed as \(\tan\), is the ratio of the sine function to the cosine function, or equivalently, the ratio of the opposite side to the adjacent side in a right-angled triangle. The function is undefined when the cosine of an angle is zero, leading to vertical asymptotes in its graph at odd multiples of \(\frac\pi2\).
Even though the original exercise does not require the tangent function directly, understanding its relationship to the sine and cosine functions is beneficial. In more complex trigonometry problems, recognizing when to use the tangent function can simplify problems, for instance, by transforming an expression into a single \(\tan\) of an angle rather than dealing with separate sine and cosine terms.
Even though the original exercise does not require the tangent function directly, understanding its relationship to the sine and cosine functions is beneficial. In more complex trigonometry problems, recognizing when to use the tangent function can simplify problems, for instance, by transforming an expression into a single \(\tan\) of an angle rather than dealing with separate sine and cosine terms.
Other exercises in this chapter
Problem 28
In Exercises 9-50, verify the identity \( \dfrac{1}{\sin x} - \dfrac{1}{\csc x} = \csc x - \sin x \)
View solution Problem 29
In Exercises 29-36, use a double-angle formula to rewrite the expression. \( 6 \sin x \cos x \)
View solution Problem 29
In Exercises 25-38, find all solutions of the equation in the interval \( [0, 2\pi) \). \( \sec^2 x - \sec x = 2 \)
View solution Problem 29
In Exercises 9-50, verify the identity \( \dfrac{1 + \sin \theta}{\cos \theta} + \dfrac{\cos \theta}{1 + \sin \theta} = 2 \sec \theta \)
View solution