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Umnica [9.8K]
3 years ago
10

Help, Help, Please

Mathematics
2 answers:
Lesechka [4]3 years ago
8 0

Answer:

θ = \frac{\pi }{3} (60° )

Step-by-step explanation:

Using the identity

sin²x + cos²x = 1 ⇒ sin²x = 1 - cos²x

Given

cos²θ - sin²θ = 2 - 5cosθ

cos²θ - (1 - cos²θ) = 2 - 5cosθ

cos²θ - 1 + cos²θ = 2 - 5cosθ

2cos²θ - 1 = 2 - 5cosθ ( subtract 2 - 5cosθ from both sides )

2cos²θ + 5cosθ - 3 = 0 ← in standard form

(cosθ + 3)(2cosθ - 1) = 0 ← in factored form

Equate each factor to zero and solve for θ

cosθ + 3 = 0

cosθ = - 3 ← not possible as - 1 ≤ cosθ  ≤ 1

2cosθ - 1 = 0

cosθ = \frac{1}{2} , so

θ = cos^{-1} (\frac{1}{2} ) = \frac{\pi }{3} ( or 60° )

pishuonlain [190]3 years ago
8 0

Answer:

\huge  \boxed{  \boxed{\blue{ { \theta = 60}^{ \circ} }}}

Step-by-step explanation:

<h3>to understand this</h3><h3>you need to know about:</h3>
  • trigonometry
  • PEMDAS
<h3>let's solve:</h3>
  1. \sf \: rewrite \:  \sin ^{2} ( \theta) \: as \: 1 -  \cos ^{2} ( \theta)  :  \\  \sf \implies \:  \cos ^{2} ( \theta)  - (1 -  \cos ^{2} ( \theta)) = 2 - 5 \cos( \theta)
  2. \sf \: remove \: parentheses \: and \: change \: its \: sign  :  \\  \sf \implies \:  \cos ^{2} ( \theta)   - 1  +   \cos ^{2} ( \theta)) = 2 - 5 \cos( \theta)
  3. \sf \: add  :  \\  \sf \implies \:  2\cos ^{2} ( \theta)   - 1  = 2 - 5 \cos( \theta)
  4. \sf \:  move \: left \: hand \: sides \: expression \: to \: right \: hand \: sides \:  :  \\  \sf \implies \:  2\cos ^{2} ( \theta)  + 5 \cos( \theta)   - 1 -2  = 0
  5. \sf \: rewrite \:  5\cos( \theta)  \: as \: 6 \cos( \theta) -  \cos( \theta)  :  \\  \sf \implies \:  2\cos ^{2} ( \theta)  + 6 \cos( \theta)    -  \cos( \theta) - 3  = 0
  6. \sf \:factor \: out \: 2 \cos( \theta)  \: and \:  - 1  :  \\  \sf \implies \:  2\cos  ( \theta)( \cos( \theta)   + 3 ) -1(  \cos( \theta)  +  3)  = 0
  7. \sf \: group:  \\  \sf \implies \:  (2\cos( \theta)  - 1) ( \cos( \theta)   + 3 )   = 0
  8. \sf \: rewrite \: as \: two \: seperate \: equation:  \\  \sf \implies \:   \begin{cases}2\cos( \theta)  - 1   = 0\\  \cos( \theta)   + 3 = 0   \end{cases}
  9. \sf add \: 1 \: to  \: the\: first \: equation \: and \: substract \: 3 \: from \: the \: second \: equation:  \\  \sf \implies \:   \begin{cases}2\cos( \theta)     = 1\\  \cos( \theta)    =  - 3 \end{cases}

\sf the \: second \: eqution \: is \: false \: \\ \sf because \: - 1  \leqslant  \cos( \theta)  \leqslant 1 \:  \\  \sf but \: we \: can \: still \: work \: with \: the \: second \: equation

  1. \sf substract \: both \: sides \: by \: 2 :  \\   \implies\frac{ 2\cos( \theta) }{2}  =  \frac{1}{2}  \\   \implies\cos( \theta)  =  \frac{1}{2}  \\   \therefore \:  \theta \:  =  {60}^{ \circ}

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hope this helps

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