Math Problem Statement
Find the exact value of the expression csc 30° * cot 45°.
Solution
The questionLet's asks to evaluate ( \csc 30^\ solvecirc \cd theot \ problemcot in the image step-by-step. The expression to evaluate is:
--- text### Step{ 1:cot } Recall Trigonometric Values45 1. **C^\osecant (\circ ( \csc \)):** \[ \$$ ### Step 1: Recall Trigonometriccsc Values \ theta = \frac{1}{\sin \-theta} \ \] (\ Attext \({ c30sc^\ }circ x \ =), \( \ \fracsin {30^\circ =1}{ \\sin xfrac}\{) 1-}{ \2}(\ \text{), so: cot } \ x[ = \ \frac{\csccos x30}{^\\circsin = \ xfrac}\{) 1Using}{ standard\ trsinigonometric values30: 1^\.circ} = \frac{1 $$\sin }{30\^\frac{circ =1 \frac{1}{2}$$, so \}{2}}(\ = 2 text \] 2{.c **scCot }angent ($$30 \^\cot$$):** circ \ = \frac{1[ }{ \\cotsin \theta30 =^\ \circfrac}{\ =cos \2theta\}{\sin \). theta2} . For \ \] (\ cot At $$4545^\^\circcirc\$$: ), $$\ \sin(\ 45sin^\ circ45 =^\ \fraccirc{\ =sqrt{2}}{ \2}$$frac and \({\ \sqrt{2}}{2}\cos )45^\circ = \frac and{\ \(\sqrtcos 45{^\2}}circ{ = \frac{\2}sqrt \{), so2}}: { \2}\), so[ : \ cot \ [ 45 ^\ \circtext = \{fraccot } {\cos 4545^\^\circcirc =}{\ \fracsin {\45cos^\ circ45}^\ =circ \frac}{{\\fracsin{\ sqrt45{^\2circ}}}{ =2}} {\1. frac \] ###{\ Step 2sqrt{2}}{: Substitute Values2 \}} =[ \1text { c \sc] }--- ###30 Step^\ circ2: Multiply \ thecd Results otNow \,text calculate{ thecot product }: \ [ 45\^\csccirc = 30^\2circ \ \cdcdotot \cot1 =45 2^\. circ =\ 2] \###cd Finalot Answer : 1The = exact value of \2(\ text\] --- {### Finalc Answersc: \ } [ \30boxed^\{circ2 \} cd\ot] Would \ youtext like{ furthercot details }, or45 do you^\ havecirc any\ questions)? is 😊 ** 2**. --- --- Would### you like Expanding your understanding a: detailed1 explanation. of Why any is of $$the \c stepssc? Here30^\circ = 2$$, are and some what related questions are to the deepen other reciprocal your understanding tr: igonometric1 functions. for How \( is 30 \(\^\circtext \{)? c2. Howsc } x\ can the) defined in unit trigon circleometry help verify? the2 values. of Why $$\ issin 45^\circ \$$ and(\ $$\cos 45sin^\ 30^\circcirc = \frac{1}{2}$$ \ a standard value? 3)? .3 What. What is the value is of $$the relationship between \ \(\cot 60cot^\ x$$, $$\sin x$$, and $$\cos x$$? 4. How do you simplify expressions involving trigonometric functions without a calculator? 5. Can you derive the trigonometric values for 45° and 30° using geometric principles? ### Tip: Memorizing standard trigonometric values for common angles (30°, 45°, 60°) can greatly speed up calculations and help in solving trigonometric problems efficiently!circ \), and how is it derived? 4. Can this method be applied for other angles like $$0^\circ$$ or $$90^\circ$$? Why or why not? 5. Why is $$\cot 45^\circ = 1$$ geometrically meaningful in a 45-45-90 triangle? --- ### Tip: Always remember that for special angles like $$30^\circ, 45^\circ,$$ and $$60^\circ$$, memorizing the sine, cosine, and tangent values is helpful in simplifying such problems!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Cosecant
Cotangent
Formulas
csc(θ) = 1 / sin(θ)
cot(θ) = cos(θ) / sin(θ)
Theorems
Reciprocal Identities in Trigonometry
Suitable Grade Level
Grades 10-12
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