Trigonometry! #8

Geometry Level 3

If tan ⁡ ( π cos ⁡ θ ) = cot ⁡ ( π sin ⁡ θ ) \tan(\pi \cos \theta) = \cot (\pi \sin \theta) and 0 < θ < 3 π 4 0 < \theta < \frac {3 \pi }{4} , then sin ⁡ ( θ + π 4 ) = ? \sin (\theta + \frac {\pi}{4}) =\ ?

This problem is part of the set Trigonometry .

1 2 2 \frac {1}{2\sqrt{2}} − 1 2 -\frac {1}{\sqrt {2}} 1 2 \frac {1}{\sqrt{2}} − 1 2 2 -\frac {1}{2\sqrt{2}}

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Omkar Kulkarni by Omkar Kulkarni , 22, India

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1 solution

Danish Ahmed
Jan 30, 2015

tan ⁡ ( π cos ⁡ θ ) = cot ⁡ ( π sin ⁡ θ ) \tan (\pi \cos \theta) = \cot (\pi \sin \theta)

So π cos ⁡ θ = π / 2 − π sin ⁡ θ \pi \cos \theta = \pi / 2 - \pi \sin \theta

or sin ⁡ θ + cos ⁡ θ = 1 / 2 \sin \theta + \cos \theta = 1 / 2

or 1 2 sin ⁡ θ + 1 2 cos ⁡ θ = 1 2 2 \frac{1}{\sqrt{2}}\sin \theta + \frac{1}{\sqrt{2}}\cos \theta = \frac{1}{2\sqrt{2}}

or sin ⁡ ( θ + π 4 ) = 1 2 2 \sin (\theta + \frac{\pi}{4}) = \frac{1}{2\sqrt{2}}

15 Helpful 0 Interesting 0 Brilliant 0 Confused

I didn't get dat how pi cos x = pi/2 - pi sin x is cos x +sin x = 1/2. Plz guide me

Chaitanya Kulkarni - 6 years, 2 months ago

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tan ⁡ ( π cos ⁡ θ ) = cot ⁡ ( π sin ⁡ θ ) = cot ⁡ ( π / 2 − π sin ⁡ θ ) \tan(\pi\cos\theta)=\cot(\pi\sin\theta)=\cot(\pi/2-\pi\sin\theta)

Danish Ahmed - 6 years, 1 month ago

Solved very neatly.! great..:-)

Anurag Pandey - 6 years, 1 month ago

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