Can The Following Equalities Hold Simultaneously Using The Basic Trigonometric Identity? 1) Sin(a) = 1 And Cos(a) = 1; 2) Sin(a) = 0 And Cos(a) = -1; 3) Sin(a) = 4/5 And Cos(a) = -3/5; 4) Sin(a) = 1/3 And Cos(a) = ?

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In the fascinating world of trigonometry, the fundamental trigonometric identity plays a pivotal role in establishing relationships between various trigonometric functions. This identity, expressed as sin²(a) + cos²(a) = 1, serves as a cornerstone for solving trigonometric equations and understanding the behavior of angles within the unit circle. In this comprehensive exploration, we will delve into the application of this identity to determine the simultaneous validity of given trigonometric equalities. By meticulously analyzing each case, we will unravel the intricacies of trigonometric relationships and enhance our grasp of this essential mathematical concept.

1) Can sin(a) = 1 and cos(a) = 1 Simultaneously Occur?

When embarking on the journey to ascertain whether sin(a) = 1 and cos(a) = 1 can coexist, we turn to the fundamental trigonometric identity for guidance. Substituting these values into the identity, we arrive at the equation 1² + 1² = 1, which simplifies to 2 = 1. This glaring contradiction immediately signals that the simultaneous occurrence of sin(a) = 1 and cos(a) = 1 is an impossibility. To further solidify this understanding, let's explore the unit circle representation of trigonometric functions. The sine of an angle corresponds to the y-coordinate of a point on the unit circle, while the cosine corresponds to the x-coordinate. The only point on the unit circle where the y-coordinate is 1 is at the angle 90 degrees (π/2 radians), and at this point, the x-coordinate (cosine) is 0, not 1. Therefore, no angle exists where both sine and cosine simultaneously equal 1, reinforcing the conclusion drawn from the trigonometric identity.

2) Can sin(a) = 0 and cos(a) = -1 Simultaneously Occur?

Now, let's shift our focus to the possibility of sin(a) = 0 and cos(a) = -1 occurring in unison. Once again, we invoke the fundamental trigonometric identity as our analytical tool. Substituting these values into the identity, we obtain 0² + (-1)² = 1, which simplifies to 1 = 1. This equation holds true, indicating that the simultaneous occurrence of sin(a) = 0 and cos(a) = -1 is indeed plausible. To gain deeper insights, we turn to the unit circle. The sine of an angle is 0 at two points on the unit circle: 0 degrees (0 radians) and 180 degrees (π radians). The cosine of an angle is -1 only at 180 degrees (π radians). Thus, the angle 180 degrees (π radians) emerges as the sole solution that satisfies both conditions simultaneously. This confirms the validity of the equality derived from the trigonometric identity and provides a visual understanding of the relationship between sine, cosine, and the unit circle.

3) Can sin(a) = 4/5 and cos(a) = -3/5 Simultaneously Occur?

Venturing further into our exploration, we now examine the scenario where sin(a) = 4/5 and cos(a) = -3/5. Employing the familiar fundamental trigonometric identity, we substitute these values to obtain (4/5)² + (-3/5)² = 1. Simplifying this equation, we get 16/25 + 9/25 = 1, which further reduces to 25/25 = 1. This equality holds true, indicating that the simultaneous occurrence of sin(a) = 4/5 and cos(a) = -3/5 is mathematically feasible. To gain a more comprehensive understanding, let's consider the implications of these values within the context of the unit circle. Since sine is positive and cosine is negative, the angle a must lie in the second quadrant, where y-coordinates are positive and x-coordinates are negative. This confirms the consistency of the given values with the established trigonometric principles and provides a geometric interpretation of the solution.

4) Can sin(a) = 1/3 and ... simultaneously occur?

To continue our investigation, we need the corresponding value for cos(a) in this scenario. Please provide the value of cos(a) to complete the analysis. Once we have the value of cos(a), we can apply the fundamental trigonometric identity and the unit circle concept, as we did in the previous cases, to determine whether the given values of sin(a) and cos(a) can simultaneously occur.

Conclusion

Through this exploration, we have witnessed the power of the fundamental trigonometric identity in discerning the compatibility of trigonometric equalities. By applying this identity and complementing it with the visual aid of the unit circle, we can navigate the intricate relationships between trigonometric functions and gain a deeper appreciation for their mathematical elegance. The fundamental trigonometric identity serves as an indispensable tool in various mathematical contexts, including solving trigonometric equations, simplifying expressions, and understanding the behavior of periodic phenomena. Its mastery is crucial for anyone seeking to excel in mathematics and related fields.

By understanding these principles, we can confidently tackle a wide array of trigonometric problems and gain a deeper understanding of the mathematical world around us. Keep exploring, keep questioning, and keep pushing the boundaries of your knowledge!