Problem 53 - Algebra 2

Question

Simplify each expression. \((\cot \theta+\tan \theta) \sin \theta\)

Step-by-Step Solution

Answer

\((\cot\theta + \tan\theta)\sin\theta = \sec\theta\)

Step 1: Identify Trigonometric Identities

We start by recalling the definitions of the trigonometric functions involved:- \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)We will use these identities to rewrite the given expression.

Step 2: Rewrite Expression Using Identities

Substitute the identities from the previous step into the expression:\((\cot \theta + \tan \theta) \sin \theta = \left(\frac{\cos \theta}{\sin \theta} + \frac{\sin \theta}{\cos \theta}\right) \sin \theta\)

Step 3: Combine Fractions

Find a common denominator for the terms in the parentheses:\(\left(\frac{\cos \theta}{\sin \theta} + \frac{\sin \theta}{\cos \theta}\right) = \frac{\cos^2 \theta + \sin^2 \theta}{\sin \theta \cos \theta}\)

Step 4: Simplify Using Pythagorean Identity

Recall the Pythagorean identity: \(\cos^2 \theta + \sin^2 \theta = 1\).Substitute this identity into the expression:\(\frac{\cos^2 \theta + \sin^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta}\)

Step 5: Multiply By \(\sin \theta\)

Now multiply the simplified fraction by \(\sin \theta\):\(\frac{1}{\sin \theta \cos \theta} \times \sin \theta = \frac{1}{\cos \theta} = \sec \theta\)

Step 6: Final Simplification

After simplifying, we find that the original expression \( (\cot \theta + \tan \theta) \sin \theta \) simplifies to \(\sec \theta\).

Key Concepts

Cotangent

The cotangent of an angle, noted as \( \cot \theta \), is one of the basic trigonometric functions. It represents the ratio of the adjacent side to the opposite side in a right-angled triangle. However, it's more commonly expressed using sine and cosine:

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

This identity stems from the definition of sine and cosine themselves.
Sine \( \sin \theta \) gives the ratio of the opposite side to the hypotenuse, while cosine \( \cos \theta \) provides the adjacent over the hypotenuse.
When using identities such as the one for cotangent in trigonometric simplifications, it allows us to transform expressions, aiding in finding simplified forms or solutions.

Tangent

The tangent, denoted as \( \tan \theta \), is another crucial function in trigonometry. It is the ratio of the opposite side to the adjacent side in a right triangle. In terms of sine and cosine, it is defined as:

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

This relationship is invaluable when simplifying or manipulating trigonometric expressions, as it allows a seamless interchange between tangent and the more fundamental sine and cosine functions.
When working with expressions involving tangent, like in this exercise, swapping it with its sine-cosine form can simplify calculations, especially when combined with other identities.

Secant

The secant function, symbolized by \( \sec \theta \), is typically less familiar than sine or cosine, but it is just as important. It is defined as the reciprocal of the cosine function:

• \( \sec \theta = \frac{1}{\cos \theta} \)

Secant becomes particularly useful when expressions have fractions involving cosines. Simplifying such fractions often leads to a direct transition from a more complex form to a simple representation using secant.
In this exercise, the final simplified form of the expression \( (\cot \theta + \tan \theta) \sin \theta \) simplifies directly to \( \sec \theta \), showcasing how secant can result in much more accessible and comprehensible expressions.

Pythagorean identity

The Pythagorean identity is one of the cornerstones of trigonometry, echoing the famous Pythagorean theorem but in trigonometric form. It states:

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

This identity is incredibly powerful for simplifying expressions, especially those involving both sine and cosine.
In the exercise, we applied this identity to the expression \( \left( \frac{\cos^2 \theta + \sin^2 \theta}{\sin \theta \cos \theta} \right) \). By replacing \( \cos^2 \theta + \sin^2 \theta \) with 1, we simplify the numerator significantly, leading to an easier path for further simplification steps.
Pythagorean identity not only simplifies expressions but often unveils hidden components, simplifying seemingly complex trigonometric tasks into manageable forms.