Algebraic Law in Geometry

Algebra Level 3

True or False:

( sin ⁡ θ ∘ × csc ⁡ θ ∘ ) + ( cos ⁡ θ ∘ × sec ⁡ θ ∘ ) + ( tan ⁡ θ ∘ × cot ⁡ θ ∘ ) (\sin \theta^{\circ} \times \csc \theta^{\circ}) +(\cos \theta^{\circ} \times \sec \theta^{\circ} )+(\tan \theta^{\circ} \times \cot \theta^{\circ})

is always equal to 3 3 .

Inspiration

I don't know True False ???

Paul Ryan Longhas by Paul Ryan Longhas , 24, Philippines

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3 solutions

Vishal S
Apr 23, 2015

If θ \theta =0 then sin ⁡ θ \sin \theta =0

Then the solution 3 does not exist for the expression

So the given statement is false

1 Helpful 0 Interesting 0 Brilliant 0 Confused

To add to the solution, the given expression is defined iff:

θ ∉ { x ∣ x = n π 2 ∀ n ∈ Z } \theta\notin \left\{x\mid x=n\frac \pi 2~\forall~n\in\Bbb Z\right\}

Prasun Biswas - 6 years ago
Chew-Seong Cheong
Apr 19, 2015

sin ⁡ θ csc ⁡ θ + cos ⁡ θ sec ⁡ θ + tan ⁡ θ cot ⁡ θ = sin ⁡ θ sin ⁡ θ + cos ⁡ θ cos ⁡ θ + tan ⁡ θ tan ⁡ θ \sin{\theta}\csc{\theta}+\cos{\theta}\sec{\theta}+ \tan{\theta}\cot{\theta} = \dfrac {\sin{\theta}}{\sin{\theta}}+\dfrac {\cos{\theta}}{\cos{\theta}}+\dfrac {\tan{\theta}}{\tan{\theta}}

The above expression is undefined when sin ⁡ θ = 0 \sin{\theta} = 0 or cos ⁡ θ = 0 \cos{\theta}=0 . Therefore, the expression is always equal to 3 3 is F a l s e \boxed{False} .

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Paul Ryan Longhas
Apr 19, 2015

If sin ⁡ θ = 0 \sin \theta = 0 or cos ⁡ θ = 0 \cos \theta = 0 , then the statement is false. So, not always equal to 3 3

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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