Problem 38 - Precalculus with Limits

Question

In Exercises 37-42, find the exact values of \( \sin 2u \), \( \cos 2u \), and \( \tan 2u \) using the double-angle formulas. \( \cos u = - \dfrac{4}{5}, \dfrac{\pi}{2} < u < \pi \)

Step-by-Step Solution

Answer

The exact values are \( \sin 2u = - \frac{24}{25} \), \( \cos 2u = \frac{7}{25} \), and \( \tan 2u = - \frac{24}{7} \)

Step 1: Compute sin(u)

We already know that \( \cos u = - \frac{4}{5} \). To find \( \sin u \) we use the Pythagorean identity \( \sin^2 u + \cos^2 u = 1 \). So, \( \sin^2 u = 1 - \cos^2 u = 1 - (- \frac{4}{5})^2 = 1 - \frac{16}{25} = \frac{9}{25} \) As \( \sin \) is positive in the second quadrant, we get \( \sin u = \frac{3}{5} \)

Step 2: Compute sin(2u)

Using the double-angle formula \( \sin 2u = 2 \sin u \cos u \), we substitute the values of \( \sin u \) and \( \cos u \) to find \( \sin 2u = 2 (\frac{3}{5})(- \frac{4}{5}) = - \frac{24}{25} \)

Step 3: Compute cos(2u)

Using the double-angle formula \( \cos 2u = \cos^2 u - \sin^2 u \), we substitute the values of \( \sin u \) and \( \cos u \) to find \( \cos 2u = (- \frac{4}{5})^2 - (\frac{3}{5})^2 = \frac{16}{25} - \frac{9}{25} = \frac{7}{25} \)

Step 4: Compute tan(2u)

We know that \( \tan u = \frac{\sin u}{\cos u} \). Therefore, \( \tan 2u = \frac{\sin 2u}{\cos 2u} = \frac{-\frac{24}{25}}{\frac{7}{25}} = - \frac{24}{7} \)

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equality are defined. One essential identity is the Pythagorean identity, given by \( \sin^2 u + \cos^2 u = 1 \). This identity helps to relate sine and cosine functions and can be used to derive other identities.

The double-angle identities are another critical set of trigonometric identities. These are especially useful when dealing with problems that involve twice an angle. The double-angle identities for sine, cosine, and tangent are:

• \( \sin 2u = 2 \sin u \cos u \)
• \( \cos 2u = \cos^2 u - \sin^2 u \)
• \( \tan 2u = \frac{2 \tan u}{1 - \tan^2 u} \)

These formulas aid in solving various trigonometric problems by simplifying expressions and breaking down complex problems into simpler parts. Knowing these identities and how to apply them is crucial for solving trigonometric equations and understanding deeper mathematical concepts.

Unit Circle

The unit circle is a powerful tool in trigonometry, and it provides a visual way to understand trigonometric functions. It is a circle with a radius of one centered at the origin of a coordinate plane. Any angle \( u \) in standard position with its vertex at the origin can be represented on the unit circle by a point \((x,y)\). Here, \( x = \cos u \) and \( y = \sin u \).

Because the unit circle allows you to neatly see the values of sine and cosine at various angles, it becomes easier to understand how these values change as the angle changes. Each point on the unit circle can be mapped to an angle, and from these points, sine, cosine, and even tangent values can be derived. This conceptual map is useful when solving problems that involve angles not typically found in standard trigonometric tables.

For this particular problem, with \( \cos u = -\frac{4}{5} \) and knowing that \( \dfrac{\pi}{2} < u < \pi \) puts us in the second quadrant, the unit circle enables us to conclude that \( \sin u \) must be positive, as previously shown in the exercise solution.

Quadrants of Trigonometric Functions

In trigonometry, the coordinate plane is divided into four quadrants, each of which affects the sign of trigonometric functions based on the angle’s terminal side. Understanding these quadrants is crucial because it allows mathematicians to determine the sign of the values such as sine and cosine for specific angles.

The four quadrants are:

• **Quadrant I**: Both sine and cosine are positive.
• **Quadrant II**: Sine is positive, and cosine is negative.
• **Quadrant III**: Both sine and cosine are negative.
• **Quadrant IV**: Sine is negative, and cosine is positive.

Knowing which quadrant an angle lies in allows you to determine the signs of the trigonometric functions for that angle. For this exercise, the angle \( u \) lies in the second quadrant, \( \frac{\pi}{2} < u < \pi \), making \( \cos u \) negative and \( \sin u \) positive. This information simplifies solving problems and ensures that calculations align with the correct trigonometric identities.