Ifsin(x)=1/3andsec(y)=17/15, Wherexandylie Between 0 And/2, Evaluate The Expression
Understanding the relationships between trigonometric functions and their values is fundamental in mathematics, especially when dealing with angles within specific intervals. In this article, we will explore the problem: If sin(x) = 1/3 and sec(y) = 17/15, where x and y are between 0 and 2, evaluate the given expression. We will break down the process step-by-step, covering the necessary concepts, calculations, and reasoning to arrive at the solution.
Understanding the Problem
Before diving into calculations, it's important to comprehend what is given and what needs to be found:
- sin(x) = 1/3, where x ∈ [0, 2] (assuming radians)
- sec(y) = 17/15, where y ∈ [0, 2]
- The goal is to evaluate a certain trigonometric expression involving x and y (the expression will be specified later).
Note: Since the original problem does not specify the exact expression to evaluate, we will assume it involves common trigonometric functions of x and y, such as sin, cos, tan, or combinations thereof. For demonstration, we will evaluate the expression: sin(x) + cos(y), which is a typical example in such problems.
Step 1: Clarify the Domain and Functions
Understanding the Domain of x and y
- Both x and y are between 0 and 2. Usually, in trigonometry, angles are measured in radians unless specified otherwise.
- The interval [0, 2] radians approximately corresponds to [0°, 114.59°], since 2 radians ≈ 114.59°.
- This range covers parts of the first quadrant and possibly into the second, depending on the specific context.
Given Values and Their Implications
- sin(x) = 1/3 ≈ 0.3333. Since sin(x) is positive in the first and second quadrants, x could be in [0, π] (or [0, 3.1416]) — but constrained further by the domain.
- sec(y) = 17/15 ≈ 1.1333. Since secant is 1/cosine, and secant is positive here, y is in quadrants where cosine is positive (quadrant I and IV).
Step 2: Find the Corresponding Cosine and Sine Values
Calculating cos(x)
Given sin(x) = 1/3, we can find cos(x) using the Pythagorean identity:
\[
\sin^2 x + \cos^2 x = 1
\]
\[
\left(\frac{1}{3}\right)^2 + \cos^2 x = 1
\]
\[
\frac{1}{9} + \cos^2 x = 1
\]
\[
\cos^2 x = 1 - \frac{1}{9} = \frac{8}{9}
\]
\[
\cos x = \pm \sqrt{\frac{8}{9}} = \pm \frac{\sqrt{8}}{3} = \pm \frac{2\sqrt{2}}{3}
\]
Since x is between 0 and 2 radians (~0° to 114.59°), and sin(x) > 0, cos(x) is positive in the first quadrant and negative in the second quadrant.
- For x in [0, π/2], cos(x) > 0
- For x in [π/2, π], cos(x) < 0
Given the domain [0, 2], which includes angles up to approximately 114.59°, x could be in either the first or second quadrant.
However, without additional information, we often consider the principal value in the first quadrant for simplicity unless stated otherwise.
Thus, the most straightforward assumption is:
\[
\boxed{
\cos x = \frac{2\sqrt{2}}{3}
}
\]
---
Calculating cos(y) from sec(y) = 17/15
Since sec(y) = 1 / cos(y):
\[
\cos y = \frac{1}{\sec y} = \frac{15}{17}
\]
Because sec(y) > 0, and y is between 0 and 2 radians, y is in a quadrant where cosine is positive — namely, quadrants I or IV.
Now, find sin(y) using the Pythagorean identity:
\[
\sin^2 y + \cos^2 y = 1
\]
\[
\sin^2 y + \left(\frac{15}{17}\right)^2 = 1
\]
\[
\sin^2 y + \frac{225}{289} = 1
\]
\[
\sin^2 y = 1 - \frac{225}{289} = \frac{289}{289} - \frac{225}{289} = \frac{64}{289}
\]
\[
\sin y = \pm \frac{8}{17}
\]
Again, considering y in [0, 2], and since sec(y) > 0, y could be in quadrants I or IV.
- In quadrant I, sin(y) > 0 → sin y = +8/17
- In quadrant IV, sin(y) < 0 → sin y = -8/17
Most problems assume the principal value in quadrant I unless otherwise specified. So, for simplicity, let:
\[
\sin y = \frac{8}{17}
\]
---
Step 3: Evaluate the Target Expression
Assuming the expression to evaluate is sin(x) + cos(y), substituting the calculated values:
\[
\sin(x) + \cos y = \frac{1}{3} + \frac{15}{17}
\]
Find a common denominator:
\[
\frac{1}{3} = \frac{17}{51}
\]
\[
\frac{15}{17} = \frac{51}{51} \times \frac{15}{17} = \frac{45}{51}
\]
Actually, to combine directly:
\[
\frac{1}{3} + \frac{15}{17} = \frac{17}{51} + \frac{45}{51} = \frac{62}{51}
\]
So, the value of the expression:
\[
\boxed{
\sin(x) + \cos y = \frac{62}{51}
}
\]
which is approximately 1.2157.
---
Additional Insights and Applications
Why Understanding These Calculations Matters
- Trigonometric identities are fundamental in fields such as engineering, physics, and computer science.
- They help solve real-world problems involving angles and distances, such as in navigation, astronomy, and signal processing.
- Being able to determine unknown angles or function values from given data is a key skill in problem-solving.
Extending the Problem to Other Expressions
- Similar steps can be used to evaluate other expressions such as tan(x) - csc(y), sin^2 x + cos^2 y, or more complex combinations.
- For example, to find tan(x):
- Similarly, csc(y):
---
Conclusion
In this comprehensive analysis, we examined the given trigonometric conditions:
- sin(x) = 1/3
- sec(y) = 17/15
and determined the corresponding values of cos(x), sin(y), and cos(y). Using these, we evaluated the sum sin(x) + cos(y), arriving at a final value of 62/51.
This process illustrates the importance of understanding fundamental identities and the careful consideration of the domain of angles. Mastery of such techniques enables solving a wide range of problems in trigonometry and related fields, emphasizing the importance of step-by-step reasoning and attention to detail in mathematical problem-solving.
If you encounter similar problems, remember to:
- Clarify the domain and quadrant considerations.
- Use Pythagorean identities to find unknown functions.
- Substitute known values carefully.
- Simplify expressions with common denominators for clarity.
By practicing these steps, you'll strengthen your trigonometric problem-solving skills and deepen your understanding of the relationships between different functions.
