Problem 12 - A Graphical Approach to Precalculus with Limits

Question

Use identities to find (a) \(\sin 2 \theta\) and (b) \(\cos 2 \theta\) $$\cos \theta=-\frac{12}{13} \text { and } \sin \theta>0$$

Step-by-Step Solution

Answer

\(\sin 2\theta = -\frac{120}{169}\), \(\cos 2\theta = \frac{119}{169}\).

Step 1: Understand Given Conditions

We are given that \(\cos \theta = -\frac{12}{13}\) and \(\sin \theta > 0\). This indicates that \(\theta\) is in the second quadrant, as cosine is negative and sine is positive there.

Step 2: Use Pythagorean Identity

The Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) can be used to find \(\sin \theta\). Plug in the value of \(\cos \theta\):\[sin^2 \theta + \left(-\frac{12}{13}\right)^2 = 1\]\[sin^2 \theta + \frac{144}{169} = 1\]Solve for \(\sin^2 \theta\):\[sin^2 \theta = 1 - \frac{144}{169} = \frac{25}{169}\]So, \(\sin \theta = \frac{5}{13}\) (since \(\sin \theta > 0\)).

Step 3: Calculate \(\sin 2\theta\) Using Double Angle Identity

The identity for \(\sin 2\theta\) is \(2 \sin \theta \cos \theta\). Substitute the values for \(\sin \theta\) and \(\cos \theta\):\[\sin 2\theta = 2 \times \frac{5}{13} \times \left(-\frac{12}{13}\right)\]\[\sin 2\theta = 2 \times \left(-\frac{60}{169}\right) = -\frac{120}{169}\].

Step 4: Calculate \(\cos 2\theta\) Using Double Angle Identities

The identity for \(\cos 2\theta\) can be \(\cos^2 \theta - \sin^2 \theta\). Use the values found:\[\cos 2\theta = \left(-\frac{12}{13}\right)^2 - \left(\frac{5}{13}\right)^2\]\[\cos 2\theta = \frac{144}{169} - \frac{25}{169}\]\[\cos 2\theta = \frac{119}{169}\].

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental relation in trigonometry that tells us about the relationship between the sine and cosine of an angle. It states that for any angle \(\theta\), the sum of the square of the sine and the square of the cosine is always equal to 1:\[\sin^2 \theta + \cos^2 \theta = 1\]This identity is extremely useful when you need to find one trigonometric function when you already know the other. For example, in the original problem, we knew \(\cos \theta = -\frac{12}{13}\) and needed to find \(\sin \theta\). Using the Pythagorean Identity, we substituted the known value:- \[\sin^2 \theta + \left(-\frac{12}{13}\right)^2 = 1\]- Solving for \(\sin^2 \theta\), we found it to be \(\frac{25}{169}\).- Hence, \(\sin \theta = \frac{5}{13}\) considering the positive scenario in the second quadrant, where sine is positive.In summary, remember:- Pythagorean Identity: Useful for switching between sine and cosine.- Helps solve trigonometric equations by providing a relation between these functions.

Double Angle Formulas

Double Angle Formulas are identities in trigonometry that express trigonometric functions of double angles, such as \(2\theta\), in terms of functions of single angles. For sine and cosine, they are given by:- \(\sin 2\theta = 2 \sin \theta \cos \theta\)- \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\)These formulas are particularly helpful when solving problems involving angles that are double the given angle. In the original exercise, they allowed us to find \(\sin 2\theta\) and \(\cos 2\theta\) using the known values \(\sin \theta\) and \(\cos \theta\):For \(\sin 2\theta\):

• Start with the formula: \(\sin 2\theta = 2 \sin \theta \cos \theta\).
• Substitute \(\sin \theta = \frac{5}{13}\) and \(\cos \theta = -\frac{12}{13}\).
• This yields \(\sin 2\theta = -\frac{120}{169}\).

For \(\cos 2\theta\):

• Begin with \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\).
• Substitute the known squares: \(\cos^2\theta = \frac{144}{169}\) and \(\sin^2\theta = \frac{25}{169}\).
• Consequently, \(\cos 2\theta = \frac{119}{169}\).

These identities simplify calculations, especially when dealing with compounded angles in trigonometry.

Trigonometric Functions

Trigonometric Functions are basic functions used to relate the angles of triangles to the lengths of their sides. The primary trigonometric functions are sine, cosine, and tangent, typically defined for angles of right triangles. Here's a refresher:- **Sine (\(\sin\))**: ratio of the length of the side of the triangle opposite the angle to the hypotenuse.- **Cosine (\(\cos\))**: ratio of the adjacent side to the hypotenuse.- **Tangent (\(\tan\))**: ratio of the opposite side to the adjacent side.In the realm of the unit circle:

• \(\sin \theta\) corresponds to the y-coordinate of the corresponding point on the unit circle.
• \(\cos \theta\) corresponds to the x-coordinate.
• The signs of these functions change depending on the quadrant in which the angle lies.

Understanding the properties of these functions helps solve complex trigonometric problems effectively. Given an angle \(\theta\), knowing whether \(\sin\), \(\cos\), and \(\tan\) are positive or negative is key:- In the first quadrant, all are positive.- In the second quadrant, only sine is positive.- In the third, only tangent.- In the fourth, only cosine.These concepts are vital in interpreting trigonometric problems and ensuring accurate solutions, as shown in our example where it was crucial to know about the behavior of sine and cosine based on the quadrant.