Oh Sine!

Geometry Level 1

Simplify: $\sqrt{1 - \sin^2 \theta}.$

\lvert \cos \theta \rvert

\lvert \sin \theta \rvert

\cos \theta

\tan \theta \

2 solutions

Martin Soliman
Dec 24, 2014

By a fundamental trigonometric identity, $1 - \sin^2 \theta = \cos^2 \theta.$ Hence, $\sqrt{1 - \sin^2 \theta} = \sqrt{\cos^2 \theta}.$ Since, by definition, $\sqrt{x^2} = \lvert x \rvert$ , then, we should have $\sqrt{1 - \sin^2 \theta} = \lvert \cos \theta \rvert.$

But cos(-x) = cosx. . So does it matter ?

Ananya Goyal - 5 years, 10 months ago

Yes, it does matter. I made the same mistake. The thing is, cos(120) would be negative but our answer can't contain any negative value. There's the need for modulus. damn

Imrul Khan - 5 years, 10 months ago

ITs not about $\cos(-x)$ being equal to $\cos(x)$ . Square root here can yield $\cos(x)$ and $-\cos(x)$ as two values so value is modules of $\cos(x)$

Ajit Deshpande - 5 years, 10 months ago

the answer wants to convey that wherever the cos will take negative value it will taken as positve because (x^2)^1/2 is always equal to |x| okkay

Sushil Kumar - 5 years, 10 months ago

Why we but the cos§ In the absolute?

Mohamed R'afat - 5 years, 8 months ago

Recall however another definition of sine and cosine, namely as linear combinations of exponential functions with an imaginary exponent.

We are not required to restrict ouselves to the real valued definition of the square root.

Alden Stowe - 5 years, 7 months ago

why is it not +/- cos(theta)...?

here it is not cos(thetha ) but numerical value is what should be considered and theta too... that is, for example: sinn(theta)= 1/root(2) and theta =135 and if you take mode cos theta how is it correct ? it's not always expressed in cos , but it is expressed numerically to get -1/root(2). here the question doen't make sense !

Arjun SivaÞrasadam - 5 years, 5 months ago

Hadia Qadir
Aug 18, 2015

1-sin^2(Ang)=Cos^2(ang),and (cos^2(ang))^1/2=+ or - cos(ang) so ans is |cos(ang)|

Oh Sine!

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