Solving Trigonometric Equations: The Case of sin40 cosx

In this article, we will explore the trigonometric equation sin40 cosx using a combination of co-function identities and fundamental trigonometric principles. We will delve into the application of these identities and provide a thorough explanation of the steps involved in solving such equations.

Understanding Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. They are useful in simplifying and solving equations involving trigonometric functions. One particular set of identities, known as the co-function identities, are essential for solving equations like sin40 cosx.

Using Co-function Identities

The co-function identity states that:

sinθ cos(90° - θ)

Using this identity, we can transform the original equation sin40 cosx into an equivalent form:

Step 1: Applying the Co-function Identity

Let's set θ 40°. According to the co-function identity:

sin40° cos(90° - 40°) cos50°

Therefore, the equation sin40 cosx can be rewritten as:

cosx cos50°

This transformation allows us to use the properties of the cosine function to find the values of x.

Solving the Equation

When we have cosx cos50°, we need to find the angles for which the cosine values are equal. There are two common scenarios to consider:

Solution 1: Direct Equality

One solution is when the angles themselves are equal:

x 50°

Solution 2: Reflection Across the Unit Circle

Another solution is when the angles are reflections across the unit circle, specifically:

x 360° - 50° 310°

These two solutions are the primary angles that satisfy the equation cosx cos50° within one full cycle of the cosine function.

Understanding the Cyclic Nature of Trigonometric Functions

Since trigonometric functions are periodic, there are infinitely many solutions for x. The general solutions for x can be expressed as:

General Solution 1

x 50° 360°n where n is any integer (for the direct equality solution).

General Solution 2

x 310° 360°n where n is any integer (for the reflection solution).

However, we can express these solutions more compactly using a single formula:

Compact General Solution

x 50° 360°n and x 310° 360°n for any integer n.

Pitfalls and Considerations

It is important to note that when dealing with trigonometric functions, the standard convention in the metric system (SI) is to interpret radians as the default unit unless specified otherwise. Therefore, if the problem statement does not specify units, the angle 40 should be interpreted as 40 radians. This interpretation might lead to different solutions than if 40 were interpreted as degrees.

For instance, if 40 were interpreted as degrees, the solutions would be different. This is a common issue in mathematics textbooks and homework problems, where the units are often ambiguous or incorrectly specified. It's crucial to pay attention to these details to avoid misunderstanding and ensure accurate solutions.

Understanding these nuances is essential for students, especially when dealing with standardized tests and exams. Always take the time to interpret the problem correctly and apply the appropriate trigonometric identities.