Is the following indentity true? sin (A) – 27 sin(A) – 3 sin2(A)-3sin(A) + 9
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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Question
![**Is the following identity true?**
\[
\frac{\sin^3(A) - 27}{\sin(A) - 3} = \sin^2(A) - 3\sin(A) + 9
\]
This text poses a mathematical question asking whether the given trigonometric identity holds true.
**Explanation:**
The expression on the left-hand side is a rational expression involving a cubic expression in the numerator \(\sin^3(A) - 27\) and a linear expression in the denominator \(\sin(A) - 3\). The right-hand side is a quadratic expression \(\sin^2(A) - 3\sin(A) + 9\).
To verify the identity, one would typically simplify the left-hand side and see if it matches the right-hand side. The cubic expression in the numerator can potentially be factored using the difference of cubes formula:
\[
a^3 - b^3 = (a-b)(a^2 + ab + b^2)
\]
In this scenario, it can be applied as:
\[
\sin^3(A) - 27 = (\sin(A) - 3)(\sin^2(A) + 3\sin(A) + 9)
\]
Dividing the entire factorization by \(\sin(A) - 3\) simplifies it to the quadratic identity on the right-hand side. Thus, the original identity is indeed true.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe069bb76-d12a-4659-9fff-a2d2132df1d6%2Fe45a6320-2cf9-48e5-b49a-39f7717d3c11%2Fhm2pv76_processed.png&w=3840&q=75)
Transcribed Image Text:**Is the following identity true?**
\[
\frac{\sin^3(A) - 27}{\sin(A) - 3} = \sin^2(A) - 3\sin(A) + 9
\]
This text poses a mathematical question asking whether the given trigonometric identity holds true.
**Explanation:**
The expression on the left-hand side is a rational expression involving a cubic expression in the numerator \(\sin^3(A) - 27\) and a linear expression in the denominator \(\sin(A) - 3\). The right-hand side is a quadratic expression \(\sin^2(A) - 3\sin(A) + 9\).
To verify the identity, one would typically simplify the left-hand side and see if it matches the right-hand side. The cubic expression in the numerator can potentially be factored using the difference of cubes formula:
\[
a^3 - b^3 = (a-b)(a^2 + ab + b^2)
\]
In this scenario, it can be applied as:
\[
\sin^3(A) - 27 = (\sin(A) - 3)(\sin^2(A) + 3\sin(A) + 9)
\]
Dividing the entire factorization by \(\sin(A) - 3\) simplifies it to the quadratic identity on the right-hand side. Thus, the original identity is indeed true.
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