Question

The versed sine and the coversed sine are defined as follows : $$\text { vers } x=1-\cos x ; \quad \text { covers } x=1-\sin x$$ $$ u=\frac{1}{\sqrt{1-k^{2} \sin ^{2} \phi}} $$

Short answer

Based on the step-by-step solution provided above, answer the following short-answer question: Q: Write the expression for u in terms of the coversed sine function. A: The expression for u in terms of the coversed sine function is given by: $$u=\frac{1}{\sqrt{1-k^2(1 - \text{covers} \phi)^2}}$$

Step-by-step solution

Write the expression using versed sine and coversed sine

The given expression is: $$u=\frac{1}{\sqrt{1-k^{2} \sin ^{2} \phi}}$$ Now, we have the definitions: $$\text { vers } x=1-\cos x ; \quad \text { covers } x=1-\sin x$$ Using these definitions, we can rewrite sin^2(φ) and cos^2(φ) in terms of versed sine and coversed sine.

Rewrite sin^2(φ) and cos^2(φ) using versed sine and coversed sine

Recall the Pythagorean identity for sine and cosine: $$\sin^2x + \cos^2x = 1$$ Since \(\text{vers} x = 1 - \cos x\), then \(\cos x = 1 - \text{vers} x\) and \(\cos^2 x = (1 - \text{vers} x)^2\). Similarly, since \(\text{covers} x = 1 - \sin x\), then \(\sin x = 1 - \text{covers} x\) and \(\sin^2 x = (1 - \text{covers} x)^2\).

Substitute sin^2(φ) and cos^2(φ) expressions to expression u

Now, let's substitute the expressions we found for sin^2(φ) and cos^2(φ) into the expression for u: $$u=\frac{1}{\sqrt{1-k^{2} \sin ^{2} \phi}} = \frac{1}{\sqrt{1-k^2(1 - \text{covers} \phi)^2}}$$

Simplify the expression

Now that we have replaced sin^2(φ) with the coversed sine function, the expression looks much simpler: $$u=\frac{1}{\sqrt{1-k^2(1 - \text{covers} \phi)^2}}$$ This is the final expression for u in terms of coversed sine function, which was our aim in solving this problem.

Key concepts

Coversed Sine

The concept of the coversed sine, often represented as \text{covers} x, is a lesser-known trigonometric function. We define \text{covers} x as:
$$\text{covers} x = 1 - \text{sin} x$$
In simple terms, it is the complement of the sine function taken from one. This function is rarely used in modern trigonometry but can be quite useful in certain mathematical and geometric contexts.

Significance in Trigonometry

The coversed sine provides an alternative way to express trigonometric identities and equations. It can often simplify complex expressions or calculations, particularly where sin²(x) appears. The relationship with the sine function is direct and it's crucial for students to understand that \text{covers} x is to sin x what \text{vers} x is to cos x—a complementary relationship.

Application in Exercises

In exercises, such as the given example, converting expressions to their coversed sine equivalents can reduce the complexity of an equation, as seen with the variable \text{u}. This approach can help students solve problems that might otherwise be daunting due to their unfamiliar appearance.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. The basic trigonometric identities include reciprocal identities, quotient identities, Pythagorean identities, co-function identities, and even-odd identities.

Building Blocks of Trigonometry

These identities are essential in simplifying trigonometric expressions and equations. They are the foundational tools that students must master in order to progress in their understanding of trigonometry.

• Reciprocal identities relate the trigonometric functions to their reciprocals.
• Quotient identities express the tangent and cotangent functions in terms of sine and cosine.
• Pythagorean identities, which include functions like the versed sine and coversed sine, are derived from the Pythagorean Theorem.
• Co-function identities establish the relationship between trigonometric functions and their complements.
• Even-odd identities denote the symmetry of trigonometric functions.

Importance in Problem Solving

Understanding these identities is not just about memorization; it's about recognizing patterns and applying these to solve various trigonometric problems effectively. For example, in the given exercise, we utilized the Pythagorean identity to aid in transforming the given equation using the coversed sine function.

Pythagorean Identity

One of the most important trigonometric identities is the Pythagorean identity, which stems from the Pythagorean Theorem concerning the relationship between the sides of a right triangle.
$$\text{sin}^2x + \text{cos}^2x = 1$$
This identity provides a direct link between the sine and cosine functions, indicating that the sum of the squares of these functions equals one.

Role in Trigonometric Functions

The Pythagorean identity is significant because it allows for the expression of one trigonometric function entirely in terms of another, which can simplify problem-solving. For instance, if we have the value of sin(x), we can find cos(x) using this identity, and vice versa.

Usage in Converting Functions

As seen in the exercise, we can rewrite \text{sin}^2(x) as \text{(1 - covers} x)^2 and \text{cos}^2(x) as \text{(1 - vers} x)^2 using the identities for coversed sine and versed sine, respectively. This reveals the underlying power of the Pythagorean identity in transforming and simplifying trigonometric expressions, a technique that can make complex problems more approachable.