Problem 18 - Algebra 2

Question

Simplify each expression. \(\cos \theta \csc \theta\)

Step-by-Step Solution

Answer

\( \cot \theta \)

Step 1: Write Down the Given Expression

The given expression to simplify is \( \cos \theta \csc \theta \). This expression involves the cosine and cosecant trigonometric functions.

Step 2: Rewrite Using Trigonometric Identities

Recall the identity for cosecant: \( \csc \theta = \frac{1}{\sin \theta} \). Substitute this identity into the expression: \( \cos \theta \cdot \frac{1}{\sin \theta} \).

Step 3: Simplify the Expression

Now simplify the expression \( \cos \theta \cdot \frac{1}{\sin \theta} \) by multiplying: \( \frac{\cos \theta}{\sin \theta} \).

Step 4: Use Trigonometric Ratio Identity

Recognize that the result from Step 3, \( \frac{\cos \theta}{\sin \theta} \), is equal to \( \cot \theta \) according to the trigonometric identity where \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Therefore, the simplified expression is \( \cot \theta \).

Key Concepts

Simplifying Expressions

Simplifying trigonometric expressions means to rewrite them in a simpler or more convenient form. It often involves using trigonometric identities to replace more complex terms with simpler ones. This process can make solving equations easier or understanding a problem clearer. For instance, when faced with an expression like \( \cos \theta \csc \theta \), the aim is to use known identities to break it down into a simpler form.

• First, identify which trigonometric functions are involved.
• Use identities that can substitute these functions to simplify the expression.
• Ultimately, you rewrite the expression using simpler terms.

In our example, simplifying expressions allows us to express \( \cos \theta \csc \theta \) as \( \cot \theta \), a more basic trigonometric function, making further calculations more straightforward.

Cosine

The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse. It plays a crucial role in various trigonometric identities and equations. Cosine is typically denoted as \( \cos \theta \), where \( \theta \) is the angle in consideration.

• It is one of the fundamental trigonometric functions.
• It helps relate angles to side lengths in triangles.
• In the unit circle, cosine corresponds to the x-coordinate.

Understanding cosine is key to simplifying expressions, as it frequently appears in many identities. For example, in the expression \( \cos \theta \csc \theta \), knowing that \( \cos \theta \) is used directly helps in manipulating the expression through identities.

Cosecant

Cosecant is the reciprocal of the sine function. It is represented as \( \csc \theta \) and is defined as \( \csc \theta = \frac{1}{\sin \theta} \). This reciprocal relationship provides a way to express sine in a different form, often useful for simplification.

• Cosecant deals with the hypotenuse and the opposite side in a right triangle.
• By understanding its identity, \( \csc \theta \) can be rewritten into more manageable terms.
• It is less commonly used but essential in simplifying expressions involving sine.

In our example expression, replacing \( \csc \theta \) with \( \frac{1}{\sin \theta} \) enables simplification, converting the original expression to \( \frac{\cos \theta}{\sin \theta} \).

Cotangent

Cotangent, another essential trigonometric function, is defined as the reciprocal of tangent. Symbolically, it is expressed as \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). It ties together the cosine and sine functions in a simple ratio.

• Cotangent represents the ratio between adjacent and opposite sides in a right triangle.
• It offers a way to merge the sine and cosine into a single expression.
• Recognized as a fractional relationship, it serves simplification goals smoothly.

Thus, in our expression \( \cos \theta \csc \theta \), simplifying results in \( \cot \theta \). This transition from a more complex ratio to a simple cotangent function highlights cotangent's utility in trigonometric simplifications.