59th Problem 2016

Geometry Level 1

cos ⁡ θ 1 + sin ⁡ θ + 1 + sin ⁡ θ cos ⁡ θ = ? \displaystyle \frac { \cos { \theta } }{ 1+\sin { \theta } } +\frac { 1+\sin { \theta } }{ \cos { \theta } } =?

Check out the set: 2016 Problems

2 cos ⁡ θ 2\cos\theta 2 sec ⁡ θ 2\sec { \theta } sec ⁡ θ \sec { \theta } 1

Angela Fajardo by Angela Fajardo , 19, Philippines

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2 solutions

Munem Shahriar
May 4, 2018

cos ⁡ θ 1 + sin ⁡ θ + 1 + sin ⁡ θ cos ⁡ θ = cos ⁡ 2 θ + ( 1 + sin ⁡ θ ) 2 ( 1 + sin ⁡ θ ) cos ⁡ θ = cos ⁡ 2 θ + sin ⁡ 2 θ + 1 + 2 sin ⁡ θ ( 1 + sin ⁡ θ ) cos ⁡ θ = 2 + 2 sin ⁡ θ ( 1 + sin ⁡ θ ) cos ⁡ θ = 2 ( 1 + sin ⁡ θ ) ( 1 + sin ⁡ θ ) cos ⁡ θ = 2 cos ⁡ θ = 2 1 sec ⁡ θ = 2 sec ⁡ θ \begin{aligned} \frac{\cos \theta}{1+ \sin \theta} + \frac{1 + \sin \theta}{\cos \theta} & = \frac{\cos^2 \theta + (1 + \sin \theta)^2}{(1 + \sin \theta)\cos \theta} \\ & = \frac{\cos^2 \theta + \sin^2 \theta + 1 + 2\sin \theta}{(1 + \sin \theta)\cos \theta} \\ & = \frac{2 + 2\sin \theta}{(1 + \sin \theta)\cos \theta}\\ & = \frac{2(1 + \sin \theta)}{(1 + \sin \theta)\cos \theta} \\ & = \frac 2{\cos \theta} \\ & = \frac 2{\frac 1{\sec \theta}} \\ & = 2 \sec \theta \end{aligned}

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Hobart Pao
Mar 14, 2016

Hi @Angela Fajardo , it's nice that you're posting a lot of problems. However, please keep in mind the following:

  • The use of \quad is really not necessary. Please do not use it.
  • Use \dfrac instead of \frac
  • Don't forget the \ in front of a trig function like \cos, \sin, \tan etc.

Thanks!

0 Helpful 0 Interesting 0 Brilliant 0 Confused

I'm sorry, I'm not really familiar with those. Can you please enlighten me?

Angela Fajardo - 5 years, 3 months ago

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When you write your LaTeX \LaTeX commands,

  • don't use \quad . they create really annoying spaces.
  • When you want to make a fraction, use \dfrac{}{} to make things look bigger and nicer
  • When you want to write sine, cosine, tangent, etc. , don't forget the \ in front like \sin, \cos, \tan, etc.

Hobart Pao - 5 years, 3 months ago

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Okay, now I understand. Thank you :D

Angela Fajardo - 5 years, 3 months ago

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