Problem 48 - Precalculus: Functions and Graphs

Question

Use fundamental Identities to write the first expression in terms of the second, for any acute angle \(\boldsymbol{\theta}\). $$\cos \theta, \cot \theta$$

Step-by-Step Solution

Answer

\( \cos \theta = \cot \theta \cdot \sqrt{1-\cot^2 \theta/(1+\cot^2 \theta)} \)

Step 1: Identify Fundamental Identities

To express \ \(\cos \theta\ \) in terms of \ \(\cot \theta\ \), start by identifying the fundamental trigonometric identities: \ \(\cot \theta = \frac{1}{\tan \theta}\ \), and \ \(\tan \theta = \frac{\sin \theta}{\cos \theta}\ \).

Step 2: Express tan theta with cot theta

Using the identity \ \(\tan \theta = \frac{1}{\cot \theta}\ \), replace \ \(\tan \theta\ \) in the formula. This gives \ \(\tan \theta = \frac{1}{\cot \theta}\ \).

Step 3: Substitute in the tan identity

Substitute \ \(\tan \theta\ \) in the identity \ \(\tan \theta = \frac{\sin \theta}{\cos \theta}\ \), which becomes \ \(\frac{1}{\cot \theta} = \frac{\sin \theta}{\cos \theta}\ \).

Step 4: Cross-multiply to isolate cos theta

By cross-multiplying \ \(\cot \theta\ \) and simplifying, we rearrange to find \ \(\cos \theta = \cot \theta \cdot \sin \theta\ \).

Step 5: Use Pythagorean Identity to find sin theta

Utilize the identity \ \(\sin^2 \theta + \cos^2 \theta = 1\ \) to express \ \(\sin \theta\ \) in terms of \ \(\cos \theta\ \), giving \ \(\sin \theta = \sqrt{1-\cos^2 \theta}\ \).

Step 6: Express cos theta in terms of cot theta

Substitute \ \(\sin \theta = \sqrt{1 - \cos^2 \theta}\ \) back into \ \(\cos \theta = \cot \theta \cdot \sin \theta\ \) to finally express \ \(\cos \theta\ \) purely in terms of \ \(\cot \theta\ \): \ \(\cos \theta = \cot \theta \cdot \sqrt{1-\cos^2 \theta}\ \).

Key Concepts

Cosine

Cosine is one of the primary trigonometric functions, often abbreviated as "cos". It belongs to the family of ratios that describe the relationships within right-angled triangles. For any given angle \( \theta \), the cosine function measures the ratio of the length of the adjacent side to the hypotenuse in a right triangle.

In a mathematical form, this can be expressed as:

• \( \cos \theta = \frac{\text{Adjacent side}}{\text{Hypotenuse}} \)

Cosine, along with sine, is key to Pythagorean identity and helps in understanding other trigonometric relations. It gets its value from the Cartesian plane when plotting the unit circle. This really tight-knit relation with sine helps fulfill various trigonometric identities.

In general, cosine's values repeat every 360 degrees (or 2\( \pi \) radians). So be sure to watch these cyclical patterns. Understanding cosine is central to utilizing trigonometric identities.

Cotangent

Cotangent, abbreviated as "cot", is another trigonometric function often derived from the fundamental function tangent. While tangent (\( \tan \theta \)) is the ratio of sine to cosine (\( \tan \theta = \frac{\sin \theta}{\cos \theta} \)), cotangent reverses this ratio:

• \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \)

Cotangent function finds its applications in trigonometry quite often, especially when dealing with reciprocal identities.

Understanding cotangent's relationship to tangent, sine, and cosine is essential. It shows how trigonometric functions can be interchanged or expressed through one another. When you need to manipulate expressions, knowing these interrelations can be very helpful.

Pythagorean Identity

Pythagorean identities form the cornerstone of various trigonometric identities and are based on the Pythagorean theorem. The most basic form is:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

These identities help express one trigonometric function in terms of another, often facilitating easier computation. For instance, if you need to solve for \( \sin \theta \) when \( \cos \theta \) is known, rearrange the identity:

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

Similarly, alternative forms are derived for tangent and cotangent:

• \( 1 + \tan^2 \theta = \sec^2 \theta \)
• \( 1 + \cot^2 \theta = \csc^2 \theta \)

These identities not only simplify complex expressions but also help tackle equations involving multiple trigonometric functions. Mastering them enables deeper understanding of trigonometric properties and their graphical significances in unit circles.