Question

How are the sine and cosine functions used to define the other four trigonometric functions?

Short answer

Question: Define the tangent, cotangent, secant, and cosecant trigonometric functions using the sine and cosine functions. Answer: The other four trigonometric functions are defined as follows: 1. Tangent: \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\) 2. Cotangent: \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} = \frac{1}{\tan(\theta)}\) 3. Secant: \(\sec(\theta) = \frac{1}{\cos(\theta)}\) 4. Cosecant: \(\csc(\theta) = \frac{1}{\sin(\theta)}\)

Step-by-step solution

Define the tangent function

The tangent function, denoted as \(\tan\), is defined in terms of the sine and cosine functions as the ratio of sine to cosine: \[\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\] where \(\theta\) is an angle in radians.

Define the cotangent function

The cotangent function, denoted as \(\cot\), is defined in terms of the sine and cosine functions as the ratio of cosine to sine. It is the reciprocal of the tangent function: \[\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} = \frac{1}{\tan(\theta)}\] where \(\theta\) is an angle in radians.

Define the secant function

The secant function, denoted as \(\sec\), is defined in terms of the cosine function as the reciprocal of cosine: \[\sec(\theta) = \frac{1}{\cos(\theta)}\] where \(\theta\) is an angle in radians.

Define the cosecant function

The cosecant function, denoted as \(\csc\), is defined in terms of the sine function as the reciprocal of sine: \[\csc(\theta) = \frac{1}{\sin(\theta)}\] where \(\theta\) is an angle in radians.

Conclusion

The other four trigonometric functions (tangent, cotangent, secant, and cosecant) are defined using the sine and cosine functions as follows: 1. \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\) 2. \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} = \frac{1}{\tan(\theta)}\) 3. \(\sec(\theta) = \frac{1}{\cos(\theta)}\) 4. \(\csc(\theta) = \frac{1}{\sin(\theta)}\) These definitions allow us to express these functions and their properties in terms of the sine and cosine functions.

Key concepts

Sine Function

The sine function is a fundamental component of trigonometry. It's used widely in mathematics, especially in the study of triangles and periodic phenomena. The sine of an angle \( \theta \) in a right-angled triangle is the ratio of the length of the side opposite the angle to the hypotenuse. This can be expressed as:

• \[\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]

This function's graph, called the sine wave, is smooth and periodic, repeating every \( 2\pi \) radians. Its range is between -1 and 1.

Cosine Function

The cosine function, integral to the world of trigonometry, complements the sine function. It represents the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. This relationship is given by:

• \[\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]

The cosine graph, known as the cosine wave, also exhibits periodic behavior with a cycle every \( 2\pi \) radians, just like the sine. The range is identical, spanning from -1 to 1.

Tangent Function

The tangent function is defined using the sine and cosine functions. Specifically, it is the ratio of the sine of an angle to its cosine:

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

Tangent in Right-Angled Triangles

Think of tangent as the ratio of the opposite side to the adjacent side in a right triangle. Its periodicity is \( \pi \) radians, differing from sine and cosine, and includes values from negative to positive infinity.

Reciprocal Functions

Reciprocal trigonometric functions involve flipping the basic sine and cosine values. These include the cosecant, secant, and cotangent functions, providing alternative perspectives.

Cosecant

- Denoted as \( \csc \), this is the reciprocal of sine: \[\csc(\theta) = \frac{1}{\sin(\theta)}\]

Secant

- Denoted as \( \sec \), this is the reciprocal of cosine: \[\sec(\theta) = \frac{1}{\cos(\theta)}\]

Cotangent

- Denoted as \( \cot \), this is the reciprocal of tangent: \[\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}\]These functions are useful for solving problems where direct ratios of sides aren't sufficient, providing flexibility in solving complex trigonometric equations.