Utilizing Sum and Difference Angle Identities to Find the Exact Value of Sin(5π/12)
Introduction to Trigonometry and Angle Identities
Trigonometry is a fundamental branch of mathematics, exploring the relationships between angles and sides of triangles. Specifically, the study of sine, cosine, and tangent plays a crucial role in various fields, including physics, engineering, and even everyday applications like navigation and design. One of the key challenges in trigonometry is finding exact values of trigonometric functions for specific angles. This task can often be simplified using angle identities, which include sum and difference identities. These identities allow us to break down complex angles into simpler ones, making it easier to find exact values.
Understanding the Sum and Difference Angle Identities
The sum and difference identities are two powerful tools in trigonometry. They are given by the following equations:
Sum Identity for Sine: [ sin(a b) sin a cos b cos a sin b ] Difference Identity for Sine: [ sin(a - b) sin a cos b - cos a sin b ]By using these identities, we can express the sine of one angle as the sine of another by breaking it down into more familiar components. In this article, we focus on how to apply these identities to find the exact value of (sinleft(frac{5pi}{12}right)).
Deriving the Value of (sinleft(frac{5pi}{12}right))
The angle (frac{5pi}{12}) can be expressed as the difference of two simpler angles:
[ frac{5pi}{12} frac{pi}{2} - frac{pi}{12} ]
Using the difference identity for sine, we can write:
[ sinleft(frac{5pi}{12}right) sinleft(frac{pi}{2} - frac{pi}{12}right) ]
Applying the difference identity, we get:
[ sinleft(frac{5pi}{12}right) sinleft(frac{pi}{2}right) cosleft(frac{pi}{12}right) - cosleft(frac{pi}{2}right) sinleft(frac{pi}{12}right) ]
Since (sinleft(frac{pi}{2}right) 1) and (cosleft(frac{pi}{2}right) 0), it simplifies to:
[ sinleft(frac{5pi}{12}right) 1 cdot cosleft(frac{pi}{12}right) - 0 cdot sinleft(frac{pi}{12}right) ]
[ sinleft(frac{5pi}{12}right) cosleft(frac{pi}{12}right) ]
Breaking Down (cosleft(frac{pi}{12}right))
Next, we can express (frac{pi}{12}) as a difference of two angles:
[ frac{pi}{12} frac{pi}{4} - frac{pi}{6} ]
Using the difference identity for cosine, we can write:
[ cosleft(frac{pi}{12}right) cosleft(frac{pi}{4} - frac{pi}{6}right) ]
Applying the difference identity for cosine, we get:
[ cosleft(frac{pi}{12}right) cosleft(frac{pi}{4}right) cosleft(frac{pi}{6}right) sinleft(frac{pi}{4}right) sinleft(frac{pi}{6}right) ]
Substituting known values, we find:
[ cosleft(frac{pi}{4}right) frac{sqrt{2}}{2}, quad cosleft(frac{pi}{6}right) frac{sqrt{3}}{2}, quad sinleft(frac{pi}{4}right) frac{sqrt{2}}{2}, quad sinleft(frac{pi}{6}right) frac{1}{2} ]
[ cosleft(frac{pi}{12}right) left(frac{sqrt{2}}{2}right) left(frac{sqrt{3}}{2}right) left(frac{sqrt{2}}{2}right) left(frac{1}{2}right) ]
[ cosleft(frac{pi}{12}right) frac{sqrt{6}}{4} frac{sqrt{2}}{4} ]
[ cosleft(frac{pi}{12}right) frac{sqrt{6} sqrt{2}}{4} ]
Therefore, (sinleft(frac{5pi}{12}right) frac{sqrt{6} sqrt{2}}{4}).
Conclusion
In conclusion, by utilizing sum and difference angle identities, we were able to find the exact value of (sinleft(frac{5pi}{12}right)). This demonstration shows the power and utility of these identities in simplifying and solving trigonometric problems. Understanding these identities and their applications is crucial for anyone studying trigonometry and related fields.
Related Keywords
sum and difference angle identities trigonometry exact values of sine anglesFrequently Asked Questions (FAQ)
Q: Why are sum and difference identities useful in trigonometry?
A: Sum and difference identities are useful because they allow us to break down complex angles into simpler ones, making it easier to find exact values of trigonometric functions. They are particularly helpful in solving problems that involve angles that are not part of the standard trigonometric tables.
Q: Can you provide an example of another angle identity problem?
A: Certainly! Let's consider finding (cosleft(frac{7pi}{12}right)). Using the sum identity for cosine, we would express (frac{7pi}{12}) as (frac{pi}{3} frac{pi}{4}) and apply the sum identity (cos(a b) cos a cos b - sin a sin b).
Q: How do these identities relate to double-angle and half-angle identities?
A: Double-angle and half-angle identities are closely related to sum and difference identities. They can be derived from these identities by setting (a b). For instance, the double-angle identity for sine ( sin(2a) 2sin a cos a ) can be derived from the sum identity (sin(a a) sin a cos a cos a sin a).
Important Notes
Remember that while these identities are powerful, they require a solid understanding of basic trigonometric values for special angles. Regular practice and thorough comprehension of these identities can greatly enhance your problem-solving skills in trigonometry.