Solving the Trigonometric Equation: sin(x) cos(14)
When faced with the equation sin(x) cos(14), it is important to understand the relationship between sine and cosine functions, particularly in the context of radians. This document will explore the solution to this equation step-by-step.
Understanding Trigonometric Functions in Radians
Trigonometric functions such as sine and cosine are typically assumed to be in radians unless a specific degree measure is explicitly indicated. Therefore, the 14 in the equation sin(x) cos(14) is understood to be in radians.
Using Trigonometric Identities
The first step is to utilize trigonometric identities to transform the equation into a more familiar form. One such useful identity is:
cos(θ) sin(π/2 - θ)
Applying this identity to cos(14) gives:
cos(14) sin(π/2 - 14)
Substitution and Solving the Equation
Substituting this result back into the original equation yields:
sin(x) sin(π/2 - 14)
Since the sine function is periodic with a period of 2π, the general solution can be expressed as:
x π/2 - 14 2kπ
where k is any integer.
Alternative Methods
Another method involves directly solving for x using the inverse sine function. The inverse sine function, commonly denoted as sin?1, can be used as follows:
x sin?1(cos(14))
The result of this calculation is:
x 76
Units of Measure
It's important to note that the calculations above are done assuming the input is in radians. If the input were to be in degrees, the calculation would yield a different result. For example, if 14 were interpreted as 14 degrees, the equation would be:
sin(x) cos(14°)
Using the identity again:
sin(76°) cos(14°)
Thus, the solution would be:
x 76°
Conclusion
In conclusion, solving the trigonometric equation sin(x) cos(14) involves understanding the relationships between sine and cosine, utilizing trigonometric identities, and considering the periodic nature of the sine function. The final solution, whether in radians or degrees, depends on the correct interpretation of the input values.