Problem 42 - 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. \( \csc u = 3, \dfrac{\pi}{2} < u < \pi \)

Step-by-Step Solution

Answer

The exact values of \( \sin 2u, \cos 2u, \tan 2u \) are \( -\frac{4\sqrt{2}}{9}, -\frac{7}{9}, \frac{4\sqrt{2}}{7} \) respectively.

Step 1: Convert cosecant to sine

Since, \(\csc u = 3 \), this implies that the \(\sin u = \frac{1}{3} \). This is because cosecant, \(\csc u \), is given as the reciprocal of the sine of \( u \) which is, \(\sin u \).

Step 2: Find cosine using Pythagorean identity

The Pythagorean relationship between sine and cosine can be expressed as \( \sin^{2}u + \cos^{2}u = 1 \). Solving for \( \cos^{2}u \), we get \( \cos^{2}u = 1 - \sin^{2}u = 1 - (\frac{1}{3})^{2} = 1 - \frac{1}{9} = \frac{8}{9} \). Since \( \frac{\pi}{2} < u < \pi \), \( u \) is in the second quadrant where cosine is negative. Therefore, \( cos u = -\sqrt{\frac{8}{9}} = -\frac{2\sqrt{2}}{3} \).

Step 3: Apply the double angle formulas

The double angle formulas for sine, cosine and tangent are given by \( \sin 2u = 2\sin u\cos u \), \( \cos 2u = \cos^{2}u -\sin^{2}u \) and \( \tan 2u = \frac{\sin 2u}{\cos 2u} \). Using these formulas, compute \( \sin 2u, \cos 2u, \tan 2u \).

Step 4: Compute the values

Substitute the values obtained in the previous steps into the formulas: \( \sin 2u = 2\sin u\cos u = 2 \times \frac{1}{3} \times -\frac{2\sqrt{2}}{3} = -\frac{4\sqrt{2}}{9} \). \( \cos 2u = \cos^{2}u -\sin^{2}u = (-\frac{2\sqrt{2}}{3})^{2} -(\frac{1}{3})^{2} =-\frac{7}{9} \). \( \tan 2u = \frac{\sin 2u}{\cos 2u} = \frac{-\frac{4\sqrt{2}}{9}}{-\frac{7}{9}}= \frac{4\sqrt{2}}{7}. \)

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental relationships between the angles and sides of a triangle in trigonometry. They help simplify complex calculations and are essential tools for trig-based exercises.
Some common trigonometric identities include:

• Pythagorean Identity: \( \sin^2 u + \cos^2 u = 1 \)
• Reciprocal Identities: \( \csc u = \frac{1}{\sin u} \), \( \sec u = \frac{1}{\cos u} \), \( \cot u = \frac{1}{\tan u} \)
• Double Angle Formulas: \( \sin 2u = 2\sin u\cos u \), \( \cos 2u = \cos^2 u - \sin^2 u \), \( \tan 2u = \frac{\sin 2u}{\cos 2u} \)

These identities allow you to transform and manipulate trigonometric expressions, which simplifies finding exact values for different angles. They are particularly handy when working with equations or verifying other trigonometric equations.

Sine

Sine is a primary trigonometric function that measures the ratio of the opposite side to the hypotenuse in a right-angled triangle. It plays a crucial role in calculating other trigonometric functions using identities.
For example, if \( \csc u = 3 \), this means that \( \sin u = \frac{1}{3} \) because cosecant is the reciprocal of sine.
Using the double angle formula for sine, \( \sin 2u = 2 \sin u \cos u \), requires finding both the sine and cosine of the original angle, \( u \).
After calculating sine from the given cosecant, the next step involves using identities like the Pythagorean Identity to find the cosine, enabling you to solve for \( \sin 2u \). This clever use of identities illustrates how interconnected these functions are.

Cosine

Cosine is another fundamental trigonometric function, showing the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
In the problem, after determining \( \sin u = \frac{1}{3} \), the Pythagorean Identity \( \sin^2 u + \cos^2 u = 1 \) helps us find \( \cos u \):
\( \cos^2 u = 1 - \left(\frac{1}{3}\right)^2 = \frac{8}{9} \). Since \( u \) is in the second quadrant \( \left(\frac{\pi}{2} < u < \pi\right) \), cosine is negative, hence \( \cos u = -\frac{2\sqrt{2}}{3} \).
The double angle formula for cosine, \( \cos 2u = \cos^2 u - \sin^2 u \), allows you to calculate \( \cos 2u \) using these foundational relationships.
This demonstrates the beauty of cosine in extending our understanding of trigonometric angles beyond just in triangles.

Tangent

Tangent is the trigonometric function derived from the ratio of sine to cosine. It is useful in expressing relationships involving angles within a triangle.
In this exercise, after finding \( \sin 2u \) and \( \cos 2u \), you can calculate \( \tan 2u \) using the formula \( \tan 2u = \frac{\sin 2u}{\cos 2u} \).
This results in \( \tan 2u = \frac{-\frac{4\sqrt{2}}{9}}{-\frac{7}{9}} = \frac{4\sqrt{2}}{7} \)

• Key insight: Tangent links sine and cosine, converting them into another valuable trigonometric expression.
• This function emphasizes how trig identities work to connect disparate angles and values, simplifying calculations.

Through these relationships, tangent enriches our toolkit when solving more complex trigonometric problems.