Problem 1 - Algebra and Trigonometry Real Mathematics, Real People

Question

Match each function with an equivalent expression. (a) \(\sin u\) (b) \(\cos u\) (c) \(\tan u\) (i) \(\frac{1}{\sec u}\) (ii) \(\frac{1}{\cot u}\) (iii) \(\frac{1}{\csc u}\)

Step-by-Step Solution

Answer

(a) matches with (iii), (b) matches with (i), and (c) matches with (ii).

Step 1: Recall the Reciprocal Identities

Recall the reciprocal identities in trigonometry: \(\sec u = \frac{1}{\cos u}\), \(\cot u = \frac{1}{\tan u}\), and \(\csc u = \frac{1}{\sin u}\)

Step 2: Match Function (a)

The function \(\sin u\) is equivalent to \(\frac{1}{\csc u}\). So, (a) corresponds with (iii). This is because \(\sin u = \frac{1}{\csc u}\).

Step 3: Match Function (b)

The function \(\cos u\) is equivalent to \(\frac{1}{\sec u}\). So, (b) corresponds with (i). The reason is \(\cos u = \frac{1}{\sec u}\).

Step 4: Match Function (c)

The function \(\tan u\) matches with \(\frac{1}{\cot u}\). So, (c) corresponds with (ii). This is because \(\tan u = \frac{1}{\cot u}\).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in the study of triangles and the properties of periodic waves. When you hear terms like

• Sine (\( \sin \)),
• Cosine (\( \cos \)),
• Tangent (\( \tan \)),
• and their reciprocals, cosecant (\( \csc \)), secant (\( \sec \)), and cotangent (\( \cot \)),

they're all interrelated and essential in trigonometry. These functions relate angles of a triangle to the ratios of its sides. Be sure to get comfortable with each function, how they're used, and their reciprocal identities, as it will help you solve various trigonometric problems. Having a good grasp of these functions is crucial for transitioning to more advanced mathematics.

Sine Function

The sine function, represented as \( \sin(u) \), is one of the main functions in trigonometry. It calculates the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.
Understanding the sine function is pivotal because it appears frequently in both simple and complex mathematics, especially in calculus and physics.

• The sine of an angle \( u \) can be defined as \( \sin(u) = \frac{\text{Opposite Side}}{\text{Hypotenuse}} \).
• In terms of reciprocal identities, \( \sin(u) \) is the reciprocal of cosecant, \( \csc(u) \), meaning \( \sin(u) = \frac{1}{\csc(u)} \).

This is essential for solving problems that include finding missing sides or angles in triangles. Remember, the sine function oscillates above and below the x-axis, highlighting its periodic nature.

Cosine Function

The cosine function, or \( \cos(u) \), is another core trigonometric function. It finds the ratio of the adjacent side to the hypotenuse in a right triangle.
This function is useful in various fields, such as engineering, physics, and even in computer graphics!

• Cosine can be expressed as \( \cos(u) = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} \).
• Its reciprocal identity is partnered with secant, \( \sec(u) \), meaning \( \cos(u) = \frac{1}{\sec(u)} \).

These relationships help in constructing graphs and solving trigonometric equations. The cosine function is periodic, repeating its values over a defined interval, and is symmetrical around the y-axis.

Tangent Function

Tangent, denoted as \( \tan(u) \), combines the properties of sine and cosine. It's the ratio of the sine of an angle to its cosine and is fundamental for expressing slopes or angles in geometry.
The importance of understanding tangent can't be overstated, particularly in calculus and real-world applications like navigation and signal processing.

• Mathematically, it’s expressed as \( \tan(u) = \frac{\sin(u)}{\cos(u)} \).
• Tangent’s reciprocal identity is cotangent, \( \cot(u) \), \( \tan(u) = \frac{1}{\cot(u)} \).

The tangent function graph presents interesting features, such as asymptotes, due to its undefined points where \( \cos(u) = 0 \). Remembering these properties will enhance your understanding of how tangent behaves and is used.