Problem 11 - 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. \(\tan 2 \theta\) \(\cot \theta=2, \theta\) lies in quadrant III.

Step-by-Step Solution

Answer

The exact values are: \(\sin 2\theta = \frac{4}{5}\), \(\cos 2\theta = -\frac{3}{5}\), and \(\tan 2\theta = -\frac{4}{3}\).

Step 1: Calculate cosine and sine

First, use the given value of cotangent, which is the reciprocal of the tangent, to compute the values of cosine and sine. Note that in the third quadrant, both sine and cosine are negative. By taking the reciprocal of the given cotangent value, \(\tan\theta = \frac{1}{2}\). This gives a 30-60-90 right triangle with the adjacent side 1 and the opposite side 2. Hence, \(\cos\theta = -\frac{1}{\sqrt{5}}\) and \(\sin\theta = -\frac{2}{\sqrt{5}}\).

Step 2: Compute double-angle trigonometric values

Use the double-angle formulae of sine, cosine and tangent functions to calculate the required values. For sine and cosine double angle, the formulae are: \(\sin 2\theta = 2\sin\theta \cos\theta\) and \(\cos 2\theta = \cos^2\theta - \sin^2\theta\)Substitute the values obtained in Step 1 into these equations: \(\sin 2\theta =2(-\frac{2}{\sqrt{5}})(-\frac{1}{\sqrt{5}}) = \frac{4}{5}\) And, \(\cos 2\theta = (-\frac{1}{\sqrt{5}})^2 - (-\frac{2}{\sqrt{5}})^2 = -\frac{3}{5}\)

Step 3: Calculate tangent double angle

Calculate tangent of the double angle, using the formula:\(\tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta}\)By substituting the values obtained in the last step, we get:\(\tan 2\theta = \frac{\frac{4}{5}}{-\frac{3}{5}} = -\frac{4}{3}\)

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in understanding angles and their relations in triangles. The basic trigonometric functions are sine (\(\sin \theta\)), cosine (\(\cos \theta\)), and tangent (\(\tan \theta\)). They are based on the ratios of the sides of a right triangle.

• Sine (\(\sin \theta\)): It is the ratio of the length of the opposite side to the hypotenuse.
• Cosine (\(\cos \theta\)): It is the ratio of the length of the adjacent side to the hypotenuse.
• Tangent (\(\tan \theta\)): It is the ratio of the length of the opposite side to the adjacent side.

These functions extend beyond triangles through the unit circle, allowing the calculation of angles beyond 0 to 90 degrees. They play a crucial role in solving problems involving angles and lengths in calculus, physics, and engineering.

Quadrant Analysis

When working with angles, understanding which quadrant an angle lies in is very important. The standard coordinate plane is divided into four quadrants, each determined by the signs of sine and cosine:

• Quadrant I: Both sine and cosine are positive.
• Quadrant II: Sine is positive, while cosine is negative.
• Quadrant III: Both sine and cosine are negative, as in our exercise.
• Quadrant IV: Cosine is positive, while sine is negative.

For angles in different quadrants, the signs of trigonometric functions change. This affects calculations and outcomes based on angles and is crucial when determining values of trigonometric identities by their location on the unit circle. In our exercise, knowing \(\theta\) is in Quadrant III helps in determining the signs of \(\sin \theta\) and \(\cos \theta\) which are both negative.

Cotangent

The cotangent function is the reciprocal of the tangent function. It is an important trigonometric function that is defined as:\[\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\]In the exercise, we start with \(\cot \theta = 2\), so \(\tan \theta = \frac{1}{2}\), indicating the ratio of the side opposite to \(\theta\) over the adjacent side in a right triangle. The cotangent helps in forming a triangle from which sine and cosine can be derived. Understanding these relationships is crucial as they provide the basis for calculating more complex trigonometric functions like sine, cosine, and tangent of double angles.

Trigonometric Identities

Trigonometric identities are formulas that allow us to express one trigonometric function in terms of others. They are essential for simplifying equations and solving trigonometric functions.Some important identities include:

• Pythagorean Identities: \[\sin^2 \theta + \cos^2 \theta = 1\]
• Double Angle Identities:
  • \[\sin 2\theta = 2\sin \theta \cos \theta\]
  • \[\cos 2\theta = \cos^2 \theta - \sin^2 \theta\]
  • \[\tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta}\]

These identities not only simplify expressions but also allow solving more complex equations that may arise in calculus or physics. They are crucial in the exercise to transform and calculate the double angle values, such as \(\sin 2\theta\), \(\cos 2\theta\), and \(\tan 2\theta\).