Problem 45
Question
In Exercises \(43-50\) , find the exact value of the trigonometric function given that \(\sin u=\frac{5}{13}\) and \(\cos v=-\frac{3}{5} .\) Both \(u\) and \(v\) are in Quadrant II.) \(\cos (u+v)\)
Step-by-Step Solution
Verified Answer
The exact value of \( \cos\left(u + v\right) \) is \( \frac{56}{65} \).
1Step 1: Determine the values of \(\cos u\) and \(\sin v\)
Since \( u \) and \( v \) are in Quadrant II, their cosine values are negative and sine values are positive. By the Pythagorean identity, \(\cos^2 u = 1 - \sin^2 u\). Thus, \(\cos u = -\sqrt{1 - \left(\frac{5}{13}\right)^2} = -\frac{12}{13}\). Similarly, \(\sin v = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \frac{4}{5}\).
2Step 2: Apply the formula for the cosine of a sum
The formula for the cosine of the sum of two angles is \(\cos\left(u + v\right) = \cos(u)\cos(v) - \sin(u)\sin(v)\). Substituting the values we found in Step 1, we get \(\cos\left(u + v\right) = \left(-\frac{12}{13}\right)\left(-\frac{3}{5}\right) - \left(\frac{5}{13}\right)\left(\frac{4}{5}\right) = \frac{56}{65}\).
Key Concepts
Cosine of a SumPythagorean IdentityExact Trigonometric Values
Cosine of a Sum
The trigonometric function cosine of a sum, represented as \( \text{cos}(u + v) \), is essential in understanding how two angles' cosines and sines interact when they are added together. According to the cosine of a sum formula, the cosine of the sum of two angles \( u \) and \( v \) is given by the expression \[ \text{cos}(u + v) = \text{cos}(u)\text{cos}(v) - \text{sin}(u)\text{sin}(v) \]. This formula is pivotal because it allows us to calculate the cosine of a new angle formed by adding two known angles without having to measure it directly.
To apply this formula effectively, remember that each trigonometric function has a corresponding value based on the angle. When dealing with exact values, this identity becomes particularly handy for demonstrating relationships between these functions without relying on approximate decimal numbers or a calculator.
For example, if you are given the sine of angle \( u \) and the cosine of angle \( v \) and need to find \( \text{cos}(u + v) \), as in the exercise, you can determine the missing values (\text{cos}(u)\
To apply this formula effectively, remember that each trigonometric function has a corresponding value based on the angle. When dealing with exact values, this identity becomes particularly handy for demonstrating relationships between these functions without relying on approximate decimal numbers or a calculator.
For example, if you are given the sine of angle \( u \) and the cosine of angle \( v \) and need to find \( \text{cos}(u + v) \), as in the exercise, you can determine the missing values (\text{cos}(u)\
Pythagorean Identity
At its core, the Pythagorean identity \( \text{cos}^2(x) + \text{sin}^2(x) = 1 \) pivots on the basic Pythagorean theorem from geometry. This identity holds a fundamental place in trigonometry, as it correlates the squares of the sine and cosine of any angle \( x \) to the value of 1. It's a powerful tool for finding unknown trigonometric values when one is known, especially within the constraints of right-angle triangles.
Moreover, when the sine or cosine is known, and you need to find the other, reorganizing the Pythagorean identity gives us \( \text{cos}^2(x) = 1 - \text{sin}^2(x) \) or \( \text{sin}^2(x) = 1 - \text{cos}^2(x) \) respectively. By taking the square root of each side, you can calculate either function's positive or negative value. It’s crucial to consider the angle’s quadrant position, as this determines the sign of the sine and cosine values.
In our exercise, we use the Pythagorean identity to find \( \text{cos}(u) \) and \( \text{sin}(v) \) because we were given \( \text{sin}(u) \) and \( \text{cos}(v) \) and knew the angles were in the second quadrant, implying specific signs for these functions.
Moreover, when the sine or cosine is known, and you need to find the other, reorganizing the Pythagorean identity gives us \( \text{cos}^2(x) = 1 - \text{sin}^2(x) \) or \( \text{sin}^2(x) = 1 - \text{cos}^2(x) \) respectively. By taking the square root of each side, you can calculate either function's positive or negative value. It’s crucial to consider the angle’s quadrant position, as this determines the sign of the sine and cosine values.
In our exercise, we use the Pythagorean identity to find \( \text{cos}(u) \) and \( \text{sin}(v) \) because we were given \( \text{sin}(u) \) and \( \text{cos}(v) \) and knew the angles were in the second quadrant, implying specific signs for these functions.
Exact Trigonometric Values
Understanding exact trigonometric values means to know the precise sine, cosine, or tangent values of specific angles without resorting to approximation. These values are typically rooted in the unit circle and right triangle definitions. For example, with familiar angles like 30°, 45°, and 60°, their trigonometric values can be exactly determined using geometric relationships.
However, the power of exact values extends beyond common angles. Anytime we can express trigonometric function values as fractions, square roots, or combinations of both, we are working with exact values. In comparison to decimal approximations, these provide a more precise and insightful understanding into the nature of the functions.
In the case of the exercise, knowing that \( \text{sin}(u) \) is \( \frac{5}{13} \) and \( \text{cos}(v) \) is \( -\frac{3}{5} \) — and then finding their complements using the Pythagorean identity — provides us with exact trigonometric values. These can then be used to find the exact value of \( \text{cos}(u + v) \) by applying the cosine of a sum identity, resulting in a precise answer without decimals.
However, the power of exact values extends beyond common angles. Anytime we can express trigonometric function values as fractions, square roots, or combinations of both, we are working with exact values. In comparison to decimal approximations, these provide a more precise and insightful understanding into the nature of the functions.
In the case of the exercise, knowing that \( \text{sin}(u) \) is \( \frac{5}{13} \) and \( \text{cos}(v) \) is \( -\frac{3}{5} \) — and then finding their complements using the Pythagorean identity — provides us with exact trigonometric values. These can then be used to find the exact value of \( \text{cos}(u + v) \) by applying the cosine of a sum identity, resulting in a precise answer without decimals.
Other exercises in this chapter
Problem 44
In Exercises 37 - 58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer. \( \dfrac{\csc \theta}{\sec
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Exercises 43-52, use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. \( \cos^4 2x \)
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In Exercises 9-50, verify the identity \( \sin t \csc \left(\dfrac{\pi}{2} - t \right) = \tan t \)
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In Exercises 37 - 58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer. \( \dfrac{1 - \sin^2 x}{\cs
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