Problem 12 - Precalculus Essentials

Question

In Exercises \(7-14,\) use the given information to find the exact value of each of the following: a. \(\sin 2 \theta\) b. \(\cos 2 \theta\) c. \(\tan 2 \theta\) \(\cot \theta=3, \theta\) lies in quadrant III.

Step-by-Step Solution

Answer

\(\sin 2 \theta = 0.6, \cos 2 \theta = 0.8, \tan 2 \theta = 0.75\).

Step 1: Finding sin and cos

Use the Pythagorean Identity to find \(\sin \theta\) and \(\cos \theta\). Given that \(\cot \theta = 3\), we know that \(\cos \theta = -3/ \sqrt{10}\) and \(\sin \theta = -1/ \sqrt{10}\) since \(\theta\) lies in the third quadrant.

Step 2: Calculating sin 2 theta

Use the double angle formula for sine, which is \(\sin 2 \theta = 2 \sin \theta \cos \theta\). Substitute the values of \(\sin \theta\) and \(\cos \theta\) to get \(\sin 2 \theta = 2 (-1/\sqrt{10}) (-3/\sqrt{10}) = 6/10 = 0.6\).

Step 3: Calculating cos 2 theta

Use the double angle formula for cosine, which is \( \cos 2 \theta = \cos^2 \theta - \sin^2 \theta\). Substitute the values of \(\sin \theta\) and \(\cos \theta\) to get \(\cos 2 \theta = (-3/\sqrt{10})^2 - (-1/\sqrt{10})^2 = 9/10 - 1/10 = 0.8\).

Step 4: Calculating tan 2 theta

Finally, use the relation \( \tan \theta = \sin \theta / \cos \theta \) to find \( \tan 2 \theta \). This yields \( \tan 2 \theta = \sin 2 \theta / \cos 2 \theta = 0.6 / 0.8 = 0.75\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables where the functions are defined. These identities are essential tools for simplifying complex trigonometric expressions and for solving trigonometric equations. The most common trigonometric identities include the reciprocal identities, quotient identities, Pythagorean identities, and co-function identities, among others.

For example, the reciprocal identities relate the sine, cosine, and tangent functions to their respective reciprocals, the cosecant, secant, and cotangent functions. Knowing these identities is crucial when solving trigonometry problems, particularly when given limited information about the angles and you need to deduce other functions, as seen in the exercise involving a cotangent value in Quadrant III.

Pythagorean Identity

The Pythagorean identity is a fundamental trigonometric identity derived from the Pythagorean theorem. It states that for any angle \( \theta \), the square of the sine plus the square of the cosine of that angle equals one: \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity is invaluable when we need to find one trigonometric function using another. For example, in the step by step solution, we deduce \( \sin \theta \) and \( \cos \theta \) using the cotangent value, leveraging the fact that the sum of their squares must equal one. Additionally, the signs of the functions are determined by the angle’s quadrant, which helps in solving problems accurately.

Double Angle Formula

The double angle formulas are trigonometric identities that relate the sine, cosine, and tangent of an angle to the sine, cosine, and tangent of twice that angle. The double angle formulas are given as follows:

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

These formulas are particularly important in trigonometry for simplifying expressions and solving equations involving periodic functions. In the given exercise, we used the double angle formulas to find the sine, cosine, and tangent of double the given angle when the cotangent and the quadrant of the original angle were provided.

Quadrant III Trigonometry

In the context of trigonometry, the coordinate plane is divided into four quadrants, and the sign of a trigonometric function depends on the quadrant in which the terminal side of the angle lies. In Quadrant III, both sine and cosine values are negative, as both the x and y coordinates are negative in this quadrant.

This information is key when working with angles in different quadrants because it affects the calculation of trigonometric functions. As demonstrated in the problem description, knowing that \( \theta \) lies in Quadrant III permits us to assign the correct signs to the sine and cosine functions, which are essential steps in calculating the double angles for \( \theta \).