Math Problem Statement

Find the exact value of each expression.

​(a)

tangent StartFraction pi Over 4 EndFractiontanπ4               

​(b) sine StartFraction pi Over 3 EndFraction

Solution

Let's computeLet's evaluate the each exact expression values step for by each of step.

the### given ( expressionsa step) by \ step:

(\tan---

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}\The) tangent ( of45 (° \ infrac degrees{\pi) can}{ be4 evaluated} using \ the) unit radians circle ( oror trigon45ometric° properties). is From a the well unit-known circle tr: igon\ometric[ value\t:

an$$left\t(\anfrac{\leftpi(}{frac4{}\piright}{)4 =}\ \right)frac{\ =sin \fracleft{\sinfrac\{\leftpi(\}{frac4{\}\piright}{4)}{\}\cosright}{\left(\cosfrac{\leftpi(}{frac4{}\piright}{)}4 =}\ \rightfrac)} {$$

frac{\Forsqrt { \2frac}}{\{pi}{2}}4{\}frac \{: sqrt$${\2}}sin{2left}}(\ =frac {\1pi}{ \4]

}\rightSo), =[ \tcosan\leftleft((\fracfrac{{\pipi}{}{44}}\right)right) = = \ frac1{
sqrt{\2]


}}###{ (2b}. )$$

(\Thussin: $$left(\tfracan{\pileft}{(\3frac}{\rightpi}{)\4) }\The sineright of) \ =(\ \fracfrac{{\pifrac}{{\3sqrt{})2 (}}60{2°}} in degrees{)frac is{\ asqrt well{-known2}} trigon{ometric2 value}} =: \ [ 1. \sin$$

leftSo(,frac {\pi[ }{\t3an}\rightleft)(\ =frac {\fracpi{}{sqrt4{}\3right}}){ =2 } 1. ]

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]

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Final### Answer : ((ab)) \ ((\ \tansin\leftleft((\fracfrac{{\pipi}{}{43}}\rightright)) = \ ):1

\The) sine
of(b ) \ \frac(\{\sinpi\}{left3(\}frac \{pi radians}{ (3or}\ right60°) =) \ isfrac another{\ standardsqrt tr{igon3ometric}} value{:

2}[ )

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sinWould\ youleft(\ like furtherfrac explanation{\ orpi additional}{ examples3?}\
rightHere) are = \5frac related{\ questionssqrt for{3 practice: }}1{2.}. What\ is]

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cos###\ *leftFinal(\ Answersfrac:{* pi(a}{)3 }\tanright\}\tanright)? left2(.frac What{\ ispi }{(\4tan}\right\left)(\ =frac {\1pi }{)
6}(bright))\ ()? \3sin.\left What is(\ \frac({\sinpi}{left3(}\fracright{)pi =}{ \6frac}{\rightsqrt){3)? 4}}.{ Find2 }(\ \cot)

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leftWould(\ youfrac like{\ furtherpi explanation}{ or4 examples}\ relatedright to)\ tr). igon5ometric values. Simpl?

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###(\ Relatedsin Questions^: 21. Whatleft(\ arefrac the cosine{\ andpi}{ tangent3 values}\ ofright ) \ +frac \{\cospi^}{32}\left \(frac radians? {\2pi}{. How3 can}\ youright use)\ the).

** unitTip circle to:** Always verify refer \ to thetan unit\ circleleft for(\ quickfrac access{\ topi tr}{igon4ometric}\right values) of = standard angles 1.? 3. What are the values of sin(π6)\sin\left(\frac{\pi}{6}\right) and tan(π6)\tan\left(\frac{\pi}{6}\right)? 4. How are the sine and cosine values related for complementary angles like π4\frac{\pi}{4} and π3\frac{\pi}{3}? 5. Can you derive the value of sin(θ)\sin(\theta) or tan(θ)\tan(\theta) using a right triangle?


Tip:

Remember the special triangle values for 3030^\circ, 4545^\circ, and 6060^\circ to quickly compute sine, cosine, and tangent values. These angles frequently appear in trigonometry problems!

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Unit Circle
Special Angles

Formulas

tan(θ) = sin(θ) / cos(θ)
Exact values for sin(π/3), cos(π/3), and tan(π/4)

Theorems

Trigonometric Ratios for Special Angles

Suitable Grade Level

Grades 9-12