Problem 37 - 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. \( \sin u = - \dfrac{3}{5}, \dfrac{3\pi}{2} < u < 2\pi \)

Step-by-Step Solution

Answer

The exact values of \( \sin 2u \), \( \cos 2u \), and \( \tan 2u \) are \( - \dfrac{24}{25} \), \( \dfrac{7}{25} \), and \( - \dfrac{24}{7} \), respectively.

Step 1: Find the value of \( \cos u \)

Using the Pythagorean identity, we substitute \( \sin u = - \dfrac{3}{5} \) into \( \sin^2 u + \cos^2 u = 1 \) to find the value of \( \cos u \). We have \( \cos u = \sqrt{1 - \sin^2 u} \). However, since \( u \) is in the fourth quadrant where cosine is positive, we get \( \cos u = \dfrac{4}{5} \).

Step 2: Find the value of \( \sin 2u \)

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

Step 3: Find the value of \( \cos 2u \)

Using the double-angle formula \( \cos 2u = \cos^2 u - \sin^2 u \), we substitute the known values of \( \sin u \) and \( \cos u \) to get \( \cos 2u = \left( \dfrac{4}{5} \right)^2 - \left( - \dfrac{3}{5} \right)^2 = \dfrac{16}{25} - \dfrac{9}{25} = \dfrac{7}{25} \).

Step 4: Find the value of \( \tan 2u \)

Using the identity \( \tan 2u = \dfrac{\sin 2u}{\cos 2u} \), we substitute the known values of \( \sin 2u \) and \( \cos 2u \) to get \( \tan 2u = \dfrac{- \dfrac{24}{25}}{\dfrac{7}{25}} = - \dfrac{24}{7} \).

Key Concepts

Double-Angle Formulas

The double-angle formulas are handy tools when dealing with trigonometric functions. They express the sine, cosine, and tangent of twice an angle in terms of the sine and cosine of the original angle. The key formulas you'll come across are:

• For sine: \( \sin(2u) = 2 \sin(u) \cos(u) \)
• For cosine: \( \cos(2u) = \cos^2(u) - \sin^2(u) \)
• For tangent: \( \tan(2u) = \frac{2 \tan(u)}{1 - \tan^2(u)} \)

These equations allow you to find the sine, cosine, and tangent of an angle that is doubled. In our exercise, we specifically used the double-angle formulas for sine and cosine to find \( \sin 2u \) and \( \cos 2u \). These formulas are instrumental for simplifying expressions and solving trigonometric equations.

Pythagorean Identity

A cornerstone of trigonometry, the Pythagorean identity, provides a relationship between the sine and cosine of an angle. The identity is:

• \( \sin^2(u) + \cos^2(u) = 1 \)

This fundamental formula allows you to solve for one trigonometric function when the other is known. In our problem, we know \( \sin u = -\frac{3}{5} \). Using this information, we substitute into the Pythagorean identity to find \( \cos u \). Solving gives us \( \cos u = \frac{4}{5} \) since \( u \) is in the fourth quadrant where cosine is positive. This identity is invaluable for finding unknown trigonometric values and ensuring all the rules of angles are followed accurately.

Sine and Cosine

Sine and cosine are two of the primary trigonometric functions. They relate to angles found within right-angled triangles. These functions have particular characteristics:

• \( \sin(\theta) \) measures the ratio of the opposite side to the hypotenuse in a right triangle.
• \( \cos(\theta) \) measures the ratio of the adjacent side to the hypotenuse.

Knowing the sign and value of these functions in each quadrant is key. In our exercise, since the angle \( u \) is located in the fourth quadrant, \( \sin u \) is negative, and \( \cos u \) is positive. This understanding helps ensure the correct application of trigonometric identities and the interpretation of function values based on the angle's location.

Trigonometric Functions

Trigonometric functions include sine, cosine, tangent, and their reciprocal functions. These functions are essential for describing the relationship between the angles and sides of triangles. They are not only limited to triangles but can also model periodic phenomena such as sound waves, light waves, and tides. Some characteristics come with having a good grasp of these functions:

• They have specific ranges and periods.
• They can be calculated using identities, like the ones used in our solution.
• Each function is related through identities such as the reciprocal and quotient identities.

The goal is to understand these relationships and apply them to different mathematical and real-world problems. In the case of our specific problem, this involved using known trigonometric functions to find exact values using double-angle and Pythagorean identities.