Question

Express each of the following in terms of

Short answer

Question: Express the given trigonometric functions in terms of sine and cosine functions: a) 3tan(θ) b) 12csc(θ) c) 5cot(θ) Answer: a) 3tan(θ) = \(\frac{\text{3}\sin(θ)}{\cos(θ)}\) b) 12csc(θ) = \(\frac{\text{12}}{\sin(θ)}\) c) 5cot(θ) = \(\frac{\text{5}\cos(θ)}{\sin(θ)}\)

Step-by-step solution

1. Definition of trigonometric functions

Recall the following trigonometric relationships: \(\tan(θ) = \frac{\sin(θ)}{\cos(θ)}\), \(\cot(θ) = \frac{\cos(θ)}{\sin(θ)}\), \( csc(θ) = \frac{1}{\sin(θ)},\) and \(sec(θ) = \frac{1}{\cos(θ)}\).

2. Express 3tan(θ) in terms of sine and cosine functions

To express 3tan(θ) in terms of sine and cosine, substitute the definition of tangent into the expression: \(\text{3tan(θ)} = \text{3}(\frac{\sin(θ)}{\cos(θ)}) = \frac{\text{3}\sin(θ)}{\cos(θ)}\)

3. Express 12csc(θ) in terms of sine and cosine functions

To express 12csc(θ) in terms of sine, substitute the definition of cosecant into the expression: \(\text{12csc(θ)} = \text{12}(\frac{1}{\sin(θ)}) = \frac{\text{12}}{\sin(θ)}\)

4. Express 5cot(θ) in terms of sine and cosine functions

To express 5cot(θ) in terms of sine and cosine, substitute the definition of cotangent into the expression: \(\text{5cot(θ)} = \text{5}(\frac{\cos(θ)}{\sin(θ)}) = \frac{\text{5}\cos(θ)}{\sin(θ)}\) In summary, the expressions in terms of sine and cosine functions are: a) 3tan(θ) = \(\frac{\text{3}\sin(θ)}{\cos(θ)}\) b) 12csc(θ) = \(\frac{\text{12}}{\sin(θ)}\) c) 5cot(θ) = \(\frac{\text{5}\cos(θ)}{\sin(θ)}\)

Key concepts

Tangent Function

Understanding the tangent function is essential for solving many trigonometric problems. The tangent of an angle, often written as \( \tan(\theta) \), is the ratio of the length of the opposing side to the adjacent side of a right-angled triangle. In terms of other trigonometric functions, it is the ratio of the sine function to the cosine function, or \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).

When we have a multiple of the tangent function, such as \( 3\tan(\theta) \), we are scaling the ratio by that multiple. This means that for every unit increase in the tangent of the angle, there is a threefold increase in the resulting value.

Sine and Cosine Relationship

Sine and cosine are fundamental trigonometric functions that represent the ratio of sides of a right triangle relative to an acute angle within the triangle. The sine of an angle \(\theta\) is the ratio of the opposite side to the hypotenuse, while the cosine is the ratio of the adjacent side to the hypotenuse.

These functions are deeply interrelated. Not only can we express tangent in terms of sine and cosine, but the Pythagorean theorem also asserts that \( \sin^2(\theta) + \cos^2(\theta) = 1 \), showcasing an intrinsic link between the two. Thus, understanding this relationship helps in expressions involving multiple trigonometric functions.

Cosecant Function

The cosecant function, written as \( \csc(\theta) \), is the reciprocal of the sine function. It is defined as \( \csc(\theta) = \frac{1}{\sin(\theta)} \). The importance of the cosecant function lies in its relationship to the sine function. While not as commonly used as sine or cosine, the cosecant provides an alternative perspective on the trigonometric ratios.

When we come across expressions like \( 12\csc(\theta) \), we are simply taking twelve times the reciprocal of the sine of angle \(\theta\). This can make calculations easier, especially when it comes to simplifying complex trigonometric expressions.

Cotangent Function

Similarly, the cotangent function, denoted as \( \cot(\theta) \), is the reciprocal of the tangent function and can also be expressed as the ratio of the cosine to the sine of an angle, \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \). It represents the ratio of the adjacent side to the opposite side in a right-angled triangle.

Expressions incorporating the cotangent, like \( 5\cot(\theta) \), involve multiplying the cotangent of an angle by a constant. Understanding the cotangent function is crucial for seeing the symmetry in trigonometry and for solving equations that may not be immediately solvable using just sine or cosine.