Problem 83 - 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 \(\cos v=1 / \sqrt{2}\). $$\cos (u-v)$$

Step-by-Step Solution

Answer

The exact value of the trigonometric expression is \(2\sqrt{2} / 5\).

Step 1: Determine the cosine value for \(u\)

Since \(u\) is in the fourth quadrant, and because \(\sin u = -3/5\), \(\cos u\) can be determined using the Pythagorean identity: \(1 = \sin^2 u + \cos^2 u\). Solving for \(\cos u\), we find that \(\cos u = \sqrt{1 - \sin^2 u} = \sqrt{1 - (-3 / 5)^2} = 4 / 5\). It is positive because cosine is positive in the fourth quadrant.

Step 2: Determine the sine value for \(v\)

Since \(v\) is in the fourth quadrant, and because \(\cos v = 1 / \sqrt{2}\), \(\sin v\) can be determined using the Pythagorean identity: \(1 = \sin^2 v + \cos^2 v\). Solving for \(\sin v\), we find that \(\sin v = \sqrt{1 - \cos^2 v} = \sqrt{1 - (1 / \sqrt{2})^2} = -1 / \sqrt{2}\). It is negative because sine is negative in the fourth quadrant.

Step 3: Determine \(\cos(u - v)\)

We substitute the values we have found into the formula for cosine of difference of two angles: \(\cos(u - v) = \cos u \cos v + \sin u \sin v = (4/5) * (1 / \sqrt{2}) + (-3/5) * (-1/ \sqrt{2}) = 2\sqrt{2} / 5\).

Key Concepts

Cosine of Angle Difference

Calculating the cosine of an angle difference is easier if you know the special trigonometric identity for this calculation. The formula is essential when working with two angles, like in our given exercise. It is expressed as:

• \( \cos(u - v) = \cos u \cos v + \sin u \sin v \)

This identity helps in breaking down the angle difference into moves with known cosine and sine values of each angle separately. This means you do not have to visualize the angles together to solve them; you use known values of each part.
For example, if you know the cosine and sine values of angles \( u \) and \( v \) separately, as we did from quadrants and identities, you just plug them into the formula to easily find \( \cos(u - v) \). Using this approach simplifies what might otherwise be a complex return to a surprisingly manageable one.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship in trigonometry. It helps us find unknown trigonometric values when one is known. The identity states:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This means that for any angle \( \theta \), the sum of the square of its sine and the square of its cosine is always equal to 1.
If you have sine but need the cosine, simply rearrange the identity to \( \cos^2 \theta = 1 - \sin^2 \theta \). After calculating \( \cos^2 \theta \), take the square root to get \( \cos \theta \), remembering to consider the sign based on the quadrant.
The identity not only guides calculations but also gives insight into the inherent balance between sine and cosine. In our problem, it directly helps find \( \cos u \) and \( \sin v \), employing known \( \sin u \) and \( \cos v \), based on their positioning in the coordinate plane.

Trigonometric Quadrants

Understanding trigonometric quadrants is crucial for determining the signs of trigonometric functions like sine, cosine, and tangent. The coordinate plane is divided into four quadrants:

• Quadrant I: All (sine, cosine, tangent) are positive.
• Quadrant II: Sine is positive; cosine and tangent are negative.
• Quadrant III: Tangent is positive; sine and cosine are negative.
• Quadrant IV: Cosine is positive; sine and tangent are negative.

Knowing which quadrant an angle belongs to helps predict the sign of its trigonometric ratios without any direct calculation. For example, in our exercise, both angles \( u \) and \( v \) were in the fourth quadrant.
This informed us that \( \cos u \) and \( \cos v \) are positive, while \( \sin u \) and \( \sin v \) are negative. Such knowledge significantly simplifies solving trigonometric expressions involving multiple angles.