Problem 14 - A Graphical Approach to Precalculus with Limits

Question

Use identities to find (a) \(\sin 2 \theta\) and (b) \(\cos 2 \theta\) $$\tan \theta=\frac{5}{3} \text { and } \sin \theta<0$$

Step-by-Step Solution

Answer

\(\sin 2\theta = -\frac{15}{17}\) and \(\cos 2\theta = -\frac{8}{17}\).

Step 1: Understand the Given Information

We know that \( \tan \theta = \frac{5}{3} \) and \( \sin \theta < 0 \). This suggests that \(\theta\) is in the fourth quadrant, where tangent is positive but sine is negative.

Step 2: Use the Pythagorean Identity to Find \(\cos \theta\)

The identity \( 1 + \tan^2 \theta = \sec^2 \theta \) can be rearranged to find \(\cos \theta\):\[ \sec^2 \theta = 1 + \left(\frac{5}{3}\right)^2 = 1 + \frac{25}{9} = \frac{34}{9} \]Thus, \( \cos^2 \theta = \frac{9}{34} \), which gives \( \cos \theta = \pm\frac{3}{\sqrt{34}} \). Since \(\theta\) is in the fourth quadrant, \( \cos \theta > 0 \), giving \( \cos \theta = \frac{3}{\sqrt{34}} \).

Step 3: Determine \(\sin \theta\)

Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), find \(\sin \theta\):\[ \sin^2 \theta = 1 - \cos^2 \theta = 1 - \frac{9}{34} = \frac{25}{34} \]Since \( \sin \theta < 0 \), \( \sin \theta = -\frac{5}{\sqrt{34}} \).

Step 4: Find \(\sin 2\theta\)

Use the double angle identity for sine: \( \sin 2\theta = 2\sin\theta\cos\theta \):\[ \sin 2\theta = 2 \left(-\frac{5}{\sqrt{34}}\right) \left(\frac{3}{\sqrt{34}}\right) = \frac{-30}{34} = -\frac{15}{17} \]

Step 5: Find \(\cos 2\theta\)

Use the double angle identity for cosine: \( \cos 2\theta = \cos^2\theta - \sin^2\theta \):\[ \cos 2\theta = \frac{9}{34} - \frac{25}{34} = -\frac{16}{34} = -\frac{8}{17} \]

Key Concepts

Pythagorean identity

The Pythagorean identity is a fundamental equation in trigonometry that relates the square of the sine and cosine of an angle. It states that for any angle \( \theta \), the following holds: \( \sin^2 \theta + \cos^2 \theta = 1 \).
This identity is named after the Pythagorean theorem because of their similarity in terms of expressing relationships in right triangles.
Using this identity, we can easily find one trigonometric function if we know the other. For example, if we know \( \cos \theta \), then \( \sin^2 \theta \) can be calculated as \( 1 - \cos^2 \theta \).

In the exercise, we first used a form of the Pythagorean identity associated with tangent and secant, given by \( 1 + \tan^2 \theta = \sec^2 \theta \), to find \( \cos \theta \).

• We calculated \( \sec^2 \theta \) and then deduced \( \cos^2 \theta \) because \( \sec \theta = \frac{1}{\cos \theta} \).
• Knowing \( \theta \) was in the fourth quadrant, we assumed \( \cos \theta \) to be positive.

Once \( \cos \theta \) was found, we used the main Pythagorean identity to find \( \sin \theta \) and confirmed it must be negative in this quadrant.

double angle identities

Double angle identities are trigonometric formulas that express trigonometric functions of double angles in terms of single angles. They are particularly useful in simplifying expressions involving angles like \( 2\theta \).
The double angle identities for sine and cosine are:

• \( \sin 2\theta = 2 \sin \theta \cos \theta \)
• \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)

These identities help in both algebraic manipulations and solving equations.

In our exercise, after finding \( \sin \theta = -\frac{5}{\sqrt{34}} \) and \( \cos \theta = \frac{3}{\sqrt{34}} \), we used these double angle identities:

• For \( \sin 2\theta \), the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \) was applied to give \( -\frac{15}{17} \).
• For \( \cos 2\theta \), the equivalent identity \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \) yielded \( -\frac{8}{17} \).

Understanding these identities is essential for dealing with problems where angles are doubled or even halved.

trigonometric functions

Trigonometric functions such as sine, cosine, and tangent are fundamental in linking angles to ratios of sides in right-angled triangles. These functions not only apply to geometry but are also essential in describing periodic phenomena.
Here's a quick overview:

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

In the unit circle, these functions can also be visualized as horizontal and vertical projections of the point on the circle corresponding to an angle \( \theta \).
In this exercise,

• We used \( \tan \theta = \frac{5}{3} \) to find the angles in a specific quadrant.
• This knowledge of quadrant hints at the signs of the trigonometric functions: in the fourth quadrant, cosine is positive while sine is negative.

Understanding these functions and how they interact with each other through identities is key to solving a wide range of trigonometric problems.