Problem 44 - Precalculus with Limits

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

Answer

The exact value of the trigonometric function \( \cos\left(u - v\right) \) is \( \dfrac{56}{65} \)

Step 1: Determine the values of \( \cos u \)

We use the Pythagorean identity \( \sin^2 u + \cos^2 u = 1 \). Since \( \sin u = \dfrac{5}{13} \), the equation becomes \( \left(\dfrac{5}{13}\right)^2 + \cos^2 u = 1 \). Solving this gives \( \cos u = - \dfrac{12}{13} \)

Step 2: Determine the values of \( \sin v \)

Just as in step 1, we use the Pythagorean identity, this time for \( v \): \( \sin^2 v + \cos^2 v = 1 \). Here, \( \cos v = -\dfrac{3}{5}\). Solving, we find \( \sin v =\dfrac{4}{5} \)

Step 3: Calculate \( \cos\left(u - v\right) \)

Substitute the values of \( \sin u, \cos u, \sin v, \cos v \) into the formula \( \cos\left(u - v\right) = \cos u \cos v + \sin u \sin v \) which gives \( \cos\left(u - v\right) = -\dfrac{12}{13}*(-\dfrac{3}{5}) + \dfrac{5}{13}*\dfrac{4}{5} \). After simplification, the final result is \( \dfrac{56}{65} \)

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry which expresses the inherent relationship between the sine and cosine of an angle. It is given by the equation:
\[ \text{sin}^2(\theta) + \text{cos}^2(\theta) = 1 \]
This identity is derived from the Pythagorean theorem applied to a right-angled triangle, where the sides are proportional to the sine and cosine of one of the non-right angles. In our problem, knowing that \(\sin u = \dfrac{5}{13}\), we used the Pythagorean identity to find \(\cos u\). Similarly, \(\cos v = - \dfrac{3}{5}\) allowed us to determine \(\sin v\) using the same identity. It’s a powerful tool because it provides a way to calculate one trigonometric function knowing another, and is crucial for solving various types of problems involving triangles and angles.

Trigonometric Function Values

Trigonometric function values are the numbers we get when we evaluate sine, cosine, tangent, and other trigonometric functions at a given angle. They describe the ratios of sides in a right triangle corresponding to the specific angle, and they extend to define the circular relationships on the unit circle. In this problem, we started with given values of \(\sin u\) and \(\cos v\) which lead us to determine \(\cos u\) and \(\sin v\) utilizing the Pythagorean identity. Here’s the key thing to remember: each trigonometric function gives us unique information about the angle or the triangle, and knowing just one value can unlock others through identities and relationships inherent in trigonometry.

Quadrant II Trigonometry

Quadrant II trigonometry refers to the trigonometric function values for angles between 90° and 180°, or between \(\pi/2\) and \(\pi\) in radians, which is where the angles \(u\) and \(v\) in our problem lie. In this quadrant, sine values are positive while cosine and tangent values are negative. The importance of understanding the sign conventions for different quadrants cannot be overstated: it guides us to the correct value of trigonometric functions for angles beyond the first quadrant. For instance, when we solved for \(\cos u\) knowing it’s in Quadrant II, we took the negative root of the square, which correctly reflects the negative cosine value in that quadrant.

Angle Difference Identity

The angle difference identity for cosine, given as \(\cos(a - b) = \cos a \cos b + \sin a \sin b\), is a powerful formula that allows us to calculate the cosine of the difference between two angles, using only the sines and cosines of the individual angles. In this exercise, we applied this identity to find \(\cos(u - v)\) after determining \(\cos u\) and \(\sin v\) separately. It's important because it broadens our ability to work with angles that are not directly given and enables us to handle complex trigonometric equations. This identity, along with the sum and difference identities for sine and tangent, are essential tools in trigonometry for breaking down more complex problems.