Sine Squared Plus Cos Squared

Trig without Tears Part 6:

The �Squared� Identities

Summary: This affiliate begins exploring trigonometric identities. 3 of them involve only squares of functions. These are called Pythagorean identities considering they�re just the good old Theorem of Pythagoras in new clothes. Acquire the actually basic one, namely sin²A + cos²A = ane, and the others are like shooting fish in a barrel to derive from information technology in a unmarried step.

Students seem to go bogged down in the huge number of trigonometric identities. Equally I said earlier, I think the problem is that students are expected to memorize all of them. But really yous don�t have to, because they�re all just forms of a very few basic identities. The adjacent couple of chapters volition explore that idea.

For example, let�southward kickoff with the actually basic identity:

(38) sin²A + cos²A = 1

That i�s easy to remember: it involves only the basic sine and cosine, and yous tin�t go the society wrong unless yous try.

right triangle, hypotenuse=1, sin A and cos A as the sides opposite and adjacent to angle A But you lot don�t have to remember even that one, since it�south really just another course of the Pythagorean Theorem. (Yous do remember that, I promise?) Just think well-nigh a correct triangle with a hypotenuse of i unit, as shown at correct.

Beginning convince yourself that the figure is right, that the lengths of the 2 legs are sinA and cosA. (Check back in the section on lengths of sides, if you need to.) At present write down the Pythagorean Theorem for this triangle. Voil�! You�ve got equation 38.

What�south overnice is that you can get the other �squared� or Pythagorean identities from this i, and you don�t have to memorize whatever of them. Only start with equation 38 and divide through by either sin²A or cos²A.

For example, what most the riddle we started with, the relation betwixt tan²A and sec²A? It�south easy to answer by a quick derivation�easier than memorizing, in my opinion.

If y'all want an identity involving tan²A, retrieve equation 3: tanA is defined to be sinA/cosA. Therefore, to create an identity involving tan²A you need sin²A/cos²A. So accept equation 38 and divide through by cos²A:

sin²A + cos²A = 1

sin²A/cos²A + cos²A/cos²A = i/cos²A

(sinA/cosA)� + 1 = (i/cosA)�

which leads immediately to the final class:

(39) tan²A + 1 = sec²A

You should be able to work out the third identity (involving cot²A and csc²A) hands enough. You can either start with equation 39 to a higher place and use the cofunction rules (equation 6 and equation seven), or start with equation 38 and split by something advisable. Either way, check to brand sure that you get

(twoscore) cot²A + ane = csc²A

It may exist easier for you to visualize these two identities geometrically. Start with the sinA, cosA, 1 right triangle above. Separate all three sides by cosA and you go the first triangle beneath; carve up by sinA instead and you become the 2d one. You can then merely read off the Pythagorean identities.

triangle with legs tan A and 1, and hypotenuse sec A triangle with legs cot A and 1, and hypotenuse csc A

From the kickoff triangle, tan²A + 1 = sec²A; from the second triangle, cot²A + i = csc²A.

Practice Issues

To get the most benefit from these issues, work them without first looking at the solutions. Refer back to the affiliate text if you need to refresh your retentiveness.

Recommendation: Work them on paper � it�s harder to fool yourself about whether yous really sympathize a problem completely.

You�ll notice full solutions for all bug. Don�t but check your answers, simply check your method too.

1 If sinA = 3/four, detect cosA.

2 tanB = −two√2. Find secB.

3 tanC = √15. Find cosC.

four tanD =√15, Find sinD.

vProve: sin²x = tan²ten / (tan²x + i)
This assumes that x ≠ π/2 + yardπ, for integer yard�or 90� + 180k�, if you adopt�because the tangent is undefined for those angles.

What�south New

20 Nov 2016: Added practice problems. Simplified the last couple of paragraphs, virtually visualizing equations 39 and xl geometrically.
1 November 2016: Updated the mathematical notation, particularly the use of italics and spaces, to conform to the standard. I used Jukka Korpela�s comprehensive Writing Mathematical Expressions (2014, Suomen E-painos Oy), ISBN 978-952-6613-25-3.
(intervening changes suppressed)
19 Feb 1997: New document.

next: vii/Sum and Difference