sin(sin" x) cofum" cos tan sin(cos"x)
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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Simplify the following. Please include steps and if there are specific rules or laws that you are using so I can learn. Thank you.
![Here is the transcription of the mathematical expressions that appear in the image:
1. Top left:
\[ \sin(\sin^{-1} x) \]
2. Top right:
\[ \cos \left( \tan^{-1} \frac{1}{3} \right) \]
3. Bottom left:
\[ \sin(\cos^{-1} x) \]
4. Bottom right:
\[ 5^{2 \log_5 x} \]
Explanation of the expressions:
1. \( \sin(\sin^{-1} x) \): This represents the sine of the arcsine (inverse sine) of \( x \). Since \( \sin^{-1} x \) is the angle whose sine is \( x \), the expression simplifies to \( x \) for values of \( x \) in the domain \([-1, 1]\).
2. \( \cos \left( \tan^{-1} \frac{1}{3} \right) \): This involves finding the cosine of the arctangent (inverse tangent) of \(\frac{1}{3}\). Here, \(\tan^{-1} \frac{1}{3}\) represents the angle whose tangent is \(\frac{1}{3}\). The cosine of this angle can be obtained using trigonometric identities or by constructing a right triangle.
3. \( \sin(\cos^{-1} x) \): This involves finding the sine of the arccosine (inverse cosine) of \( x \). Since \(\cos^{-1} x\) is the angle whose cosine is \( x \), the expression can be simplified using trigonometric identities.
4. \( 5^{2 \log_5 x} \): This expression can be simplified using properties of logarithms and exponents. \(\log_5 x\) is the logarithm of \( x \) to the base 5, and \( 2 \log_5 x \) can be interpreted as the exponent to which 5 must be raised to get \( x^2 \). Therefore, \( 5^{2 \log_5 x} = x^2 \).
Each of these mathematical expressions can be explored and expanded upon in educational settings to illustrate important concepts in trigonometry and logarithms.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc59faf40-90fa-4a45-a3f2-1ae0c0471a65%2F13af43c6-99ea-4b57-91ff-5fb8debdb140%2F8tmx0q.png&w=3840&q=75)
Transcribed Image Text:Here is the transcription of the mathematical expressions that appear in the image:
1. Top left:
\[ \sin(\sin^{-1} x) \]
2. Top right:
\[ \cos \left( \tan^{-1} \frac{1}{3} \right) \]
3. Bottom left:
\[ \sin(\cos^{-1} x) \]
4. Bottom right:
\[ 5^{2 \log_5 x} \]
Explanation of the expressions:
1. \( \sin(\sin^{-1} x) \): This represents the sine of the arcsine (inverse sine) of \( x \). Since \( \sin^{-1} x \) is the angle whose sine is \( x \), the expression simplifies to \( x \) for values of \( x \) in the domain \([-1, 1]\).
2. \( \cos \left( \tan^{-1} \frac{1}{3} \right) \): This involves finding the cosine of the arctangent (inverse tangent) of \(\frac{1}{3}\). Here, \(\tan^{-1} \frac{1}{3}\) represents the angle whose tangent is \(\frac{1}{3}\). The cosine of this angle can be obtained using trigonometric identities or by constructing a right triangle.
3. \( \sin(\cos^{-1} x) \): This involves finding the sine of the arccosine (inverse cosine) of \( x \). Since \(\cos^{-1} x\) is the angle whose cosine is \( x \), the expression can be simplified using trigonometric identities.
4. \( 5^{2 \log_5 x} \): This expression can be simplified using properties of logarithms and exponents. \(\log_5 x\) is the logarithm of \( x \) to the base 5, and \( 2 \log_5 x \) can be interpreted as the exponent to which 5 must be raised to get \( x^2 \). Therefore, \( 5^{2 \log_5 x} = x^2 \).
Each of these mathematical expressions can be explored and expanded upon in educational settings to illustrate important concepts in trigonometry and logarithms.
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