Problem 82 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact value of the trigonometric expression when \(u\) and \(v\) are in Quadrant IV and \(\sin u=-\frac{3}{5}\) and \(\cos v=1 / \sqrt{2}\). $$\sin (u-v)$$

Step-by-Step Solution

Answer

\(\sin (u - v) = -\frac{1}{5\sqrt{2}} + \frac{4\sqrt{2}}{10} = \frac{3\sqrt{2}}{10}\)

Step 1: Find the value of cos(u)

Use the Pythagorean Identity, \( \sin^2 (u) + \cos^2 (u) = 1 \). Given that \( \sin(u) = -\frac{3}{5} \), by substituting this into the Pythagorean identity, the equation will be: \((- \frac{3}{5})^2 + cos^2(u) = 1\). Simplifying that will give us \( \frac{9}{25} + cos^2(u) = 1 \). So, \(cos^2(u) = 1 - \frac{9}{25} = \frac{16}{25}\). Therefore, \(cos(u) = \sqrt{\frac{16}{25}} = \frac{4}{5}\) or \(- \frac{4}{5}\). Since u is in Quadrant IV where cosine is positive, \( cos(u) = + \frac{4}{5}\).

Step 2: Find the value of sin(v)

Similar to the previous step, utilize the Pythagorean Identity, \( \sin^2 (v) + \cos^2 (v) = 1 \). Given \( \cos(v) = \frac{\sqrt{2}}{2} \), substitute it into the Pythagorean identity, the equation will be: \( \sin^2(v) + (\frac{\sqrt{2}}{2})^2 = 1 \). Simplify that will result in \( \sin^2(v) + \frac{1}{2} = 1 \). Thus, \( \sin^2(v) = 1 - \frac{1}{2} = \frac{1}{2} \). Therefore, \( sin(v) = \sqrt{\frac{1}{2}} = \frac{\sqrt{2}}{2} \) or \(- \frac{\sqrt{2}}{2}\). Since v is in Quadrant IV where sine is negative, \( sin(v) = - \frac{\sqrt{2}}{2} \).

Step 3: Use the formula for the difference of two angles

Now, substitute the values of \(\sin (u), \cos (u), \sin (v)\), and \(\cos (v)\) into the formula for the difference of two angles \(\sin (u - v) = \sin u \cos v - \cos u \sin v\). Substituting the values found in steps 1 and 2 gives: \(\sin (u - v) = -\frac{3}{5} \times \frac{\sqrt{2}}{2} - \frac{4}{5} \times -\frac{\sqrt{2}}{2}\). Calculate to find \(\sin (u - v)\).

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry that relates the square of the sine and cosine of an angle to the value of 1. Specifically, it states that for any angle \theta, the following equation holds true:
\[\begin{equation}\boxed{\sin^2(\theta) + \cos^2(\theta) = 1}\end{equation}\]
This identity is derived from the Pythagorean theorem, which you may recall from geometry as it pertains to the relationship between the sides of a right triangle. In the context of trigonometry, the Pythagorean identity helps us find the value of one trigonometric function given another. As showcased in the exercise, knowing the value of the sine allowed us to use the identity to find the exact value of the cosine for angle u in Quadrant IV. This is crucial, as it simplifies solving various trigonometric problems, including those that involve compositions and differences of angles.

Sin and Cos in Different Quadrants

Understanding the behavior of sine and cosine functions in different quadrants of the unit circle is important for solving trigonometry problems. The unit circle is divided into four quadrants, which are distinguished by the signs (+ or -) of the sine and cosine functions. Here’s a quick recap:

• In Quadrant I, both sin and cos are positive.
• In Quadrant II, sin is positive and cos is negative.
• In Quadrant III, both sin and cos are negative.
• In Quadrant IV, sin is negative and cos is positive.

In our exercise, we needed to determine the signs of \(\sin v\) and \(\cos u\) because they were located in Quadrant IV. Knowing that the cosine of an angle in Quadrant IV is positive and the sine is negative, assisted us in finding the exact values needed to solve for \(\sin(u-v)\). Without understanding this concept, it would be challenging to proceed with the calculation and deduce the signs correctly, since only the numerical value of the sine or cosine does not convey its direction on the coordinate plane.

Angle Difference Identity for Sine

The angle difference identity for sine is an invaluable tool in trigonometry, as it enables you to find the sine of the difference between two angles based on the sines and cosines of those individual angles. The formula for the sine of the difference between two angles u and v is given by:
\[\begin{equation}\sin (u - v) = \sin u \cos v - \cos u \sin v\end{equation}\]
By using this identity, we can calculate the exact trigonometric values without having to visualize the angles or use a calculator capable of trigonometric functions. This identity is particularly helpful when the angles involved are not standard angles (those for which we readily know the sine and cosine values). In the step-by-step solution, after finding the values \(\sin u\), \(\cos u\), \(\sin v\), and \(\cos v\), we applied this identity and easily found the value of \(\sin(u-v)\) by substituting in the known values. This example illustrates how angle identities transform complex problems into simpler algebraic operations.